Research Article A Quantum Mermin-Wagner Theorem for a ...

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 637375 20 pageshttpdxdoiorg1011552013637375

Research ArticleA Quantum Mermin-Wagner Theorem fora Generalized Hubbard Model

Mark Kelbert12 and Yurii Suhov234

1 Swansea University Singleton Park Swansea SA2 8PP UK2 Instituto de Mathematica e Estatistica USP Rua de Matao 1010 Cidada Universitaria 05508-090 Sao Paulo SP Brazil3 Statistical Laboratory University of Cambridge Wilberforce Road Cambridge CB3 0WB UK4 IITP RAS Bolshoy Karetny per 18 Moscow 127994 Russia

Correspondence should be addressed to Mark Kelbert markkelbertgmailcom

Received 20 March 2013 Accepted 14 May 2013

Academic Editor Christian Maes

Copyright copy 2013 M Kelbert and Y Suhov This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs withcontinuous spins in the spirit of the Mermin-Wagner theorem In the model considered here the phase space of a single spinis H

1= L

2(119872) where 119872 is a 119889-dimensional unit torus 119872 = R119889

Z119889 with a flat metric The phase space of 119896 spins is H119896=

Lsym2

(119872

119896) the subspace of L

2(119872

119896) formed by functions symmetric under the permutations of the arguments The Fock space H

= oplus119896=01

H119896yields the phase space of a system of a varying (but finite) number of particles We associate a spaceH ≃ H(119894) with

each vertex 119894 isin Γ of a graph (ΓE) satisfying a special bidimensionality property (Physically vertex 119894 represents a heavy ldquoatomrdquoor ldquoionrdquo that does not move but attracts a number of ldquolightrdquo particles) The kinetic energy part of the Hamiltonian includes (i)minusΔ2 the minus a half of the Laplace operator on 119872 responsible for the motion of a particle while ldquotrappedrdquo by a given atomand (ii) an integral term describing possible ldquojumpsrdquo where a particle may join another atom The potential part is an operator ofmultiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials 119880(1)

(119909)119909 isin 119872 describing a field generated by a heavy atom (b) two-body potentials119880(2)

(119909 119910) 119909 119910 isin 119872 showing the interaction betweenpairs of particles belonging to the same atom and (c) two-body potentials119881(119909 119910) 119909 119910 isin 119872 scaled along the graph distance d(119894 119895)between vertices 119894 119895 isin Γ which gives the interaction between particles belonging to different atomsThe systemunder considerationcan be considered as a generalized (bosonic) Hubbard model We assume that a connected Lie group G acts on119872 represented bya Euclidean space or torus of dimension 119889

1015840le 119889 preserving the metric and the volume in 119872 Furthermore we suppose that the

potentials119880(1)119880(2) and 119881 are G-invariant The result of the paper is that any (appropriately defined) Gibbs states generated by theabove Hamiltonian is G-invariant provided that the thermodynamic variables (the fugacity 119911 and the inverse temperature 120573) satisfya certain restriction The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the densitymatrices

1 Introduction

11 Basic Facts on Bi-Dimensional Graphs As in [1] wesuppose that the graph (ΓE) has been given with the set ofvertices Γ and the set of edges E The graph has the propertythat whenever edge (119895

1015840 119895

10158401015840) isin E the reversed edge (119895

10158401015840 119895

1015840)

belongs to E as well Furthermore graph (ΓE) is without

multiple edges and has a bounded degree that is the numberof edges (119895 119895

1015840) with a fixed initial or terminal vertex is

uniformly bounded

sup [max (♯ 1198951015840 isin Γ (119895 119895

1015840) isin E

♯ 119895

1015840isin Γ (119895

1015840 119895) isin E) 119895 isin Γ] lt infin

(1)

2 Advances in Mathematical Physics

The bi-dimensionality property is expressed in the bound

0 lt sup [1119899

♯Σ (119895 119899) 119895 isin Γ 119899 = 1 2 ] lt infin (2)

where Σ(119895 119899) stands for the set of vertices in Γ at the graphdistance 119899 from 119895 isin Γ (a sphere of radius 119899 around 119895)

Σ (119895 119899) = 119895

1015840isin Γ d (119895 119895

1015840) = 119899 (3)

(The graph distance d(119895 1198951015840) = dΓE(119895 119895

1015840) between 119895 119895

1015840isin Γ is

determined as theminimal length of a path on (ΓE) joining 119895and 1198951015840)This implies that for any 119900 isin Γ the cardinality ♯Λ(119900 119899)of the ball

Λ (119900 119899) = 119895

1015840isin Γ d (119900 119895

1015840) le 119899 (4)

grows at most quadratically with 119899A justification for putting a quantum system on a graph

can be that graph-like structures become increasingly popu-lar in rigorous StatisticalMechanics for example in quantumgravity Namely see [2ndash4] On the other hand a number ofproperties of Gibbs ensembles do not depend upon ldquoregular-ityrdquo of an underlying spatial geometry

12 A BosonicModel in the Fock Space With each vertex 119894 isin Γ

we associate a copy of a compact manifold119872 which we takein this paper to be a unit 119889-dimensional torus R119889

Z119889 witha flat metric 120588 and the volume V We also associate with119894 isin Γ a bosonic Fock-Hilbert space H(119894) ≃ H Here H =

oplus119896=01

H119896whereH

119896= Lsym

2(119872

119896) is the subspace in L

2(119872

119896)

formed by functions symmetric under a permutation of thevariables Given a finite set Λ sub Γ we setH(Λ) = otimes

119894isinΛH(119894)

An element 120601 isin H(Λ) is a complex function

xlowastΛisin 119872

lowastΛ997891997888rarr 120601 (xlowast

Λ) (5)

Here xlowastΛis a collection xlowast(119895) 119895 isin Λ of finite point sets

xlowast(119895) sub 119872 associated with sites 119895 isin Λ Following [1] we callxlowast(119895) a particle configuration at site 119895 (which can be empty)and xlowast

Λa particle configuration in or over Λ The space

119872

lowastΛ of particle configurations inΛ can be represented as theCartesian product (119872lowast

)

timesΛ where 119872

lowast is the disjoint union⋃

119896=01119872

(119896) and 119872

(119896) is the collection of (unordered) 119896-point subsets of119872 (One can consider119872(119896) as the factor of theldquooff-diagonalrdquo set 119872119896

=in the Cartesian power 119872119896 under the

equivalence relation induced by the permutation group oforder 119896) The norm and the scalar product in H

Λare given

by

1003817100381710038171003817

1206011003817100381710038171003817

= (int

119872lowastΛ

1003816100381610038161003816

120601(xlowastΛ)

1003816100381610038161003816

2dxlowastΛ)

12

⟨1206011120601

2⟩ = int

119872lowastΛ

1206011(xlowast

Λ)120601

2(xlowast

Λ)dxlowast

Λ

(6)

where measure dxlowastΛis the product times

119895isinΛdxlowast(119895) and dxlowast(119895) is

the Poissonian sum measure on119872

lowast

dxlowast (119895) = sum

119896=01

1 (♯xlowast (119895) = 119896)

1

119896

prod

119909isinxlowast(119895)dV (119909) 119890minusV(119872)

(7)

Here V(119872) is the volume of torus119872As in [1] we assume that an action

(g 119909) isin G times119872 997891997888rarr g119909 isin 119872 (8)

is given of a group G that is a Euclidean space or a torus ofdimension 119889

1015840le 119889 The action is written as

g119909 = 119909 + 120579119860 mod 1 (9)

Here vector 120579 = (1205791 120579

1198891015840) with components 120579

119897isin [0 1)

and 120579119860 is the 119889-dimensional vector 120579 = ((120579119860)1 (120579119860)

119889)

representing the element g where 119860 is a (1198891015840 times 119889) matrix ofcolumn rank 1198891015840 with rational entries The action of G is liftedto unitary operators U

Λ(g) inH

Λ

UΛ(g)120601 (xlowast

Λ) = 120601 (g

minus1xlowastΛ) (10)

where gminus1xlowastΛ= gminus1xlowast(119895) 119895 isin Λ and gminus1xlowast(119895) = gminus1119909 119909 isin

xlowast(119895)The generally accepted view is that the Hubbard model

is a highly oversimplified model for strongly interactingelectrons in a solidTheHubbardmodel is a kind ofminimummodel which takes into account quantummechanicalmotionof electrons in a solid and nonlinear repulsive interactionbetween electrons There is little doubt that the model is toosimple to describe actual solids faithfully [5] In our contextthe Hubbard Hamiltonian H

Λof the system in Λ acts as

follows

(HΛ120601) (xlowast

Λ) =

[

[

minus

1

2

sum

119895isinΛ

sum

119909isinxlowast(119895)Δ

(119909)

119895+ sum

119895isinΛ

sum

119909isinxlowast(119895)119880

(1)(119909)

+

1

2

sum

119895isinΛ

sum

1199091199091015840isinxlowast(119895)

1 (119909 = 119909

1015840)119880

(2)(119909 119909

1015840)

+

1

2

sum

1198951198951015840isinΛ

1 (119895 = 119895

1015840) 119869 (d (119895 119895

1015840))

times sum

119909isinxlowast(119895)1199091015840isinxlowast(1198951015840)

119881 (119909 119909

1015840)]

]

120601 (xlowastΛ)

+ sum

1198951198951015840isinΛ

12058211989511989510158401 (♯xlowast (119895) ge 1 ♯xlowast (1198951015840) lt 120581)

times sum

119909isinxlowast(119895)int

119872

V (d119910)

times [120601 (xlowast(119895119909)rarr (1198951015840

119910)

Λ) minus 120601 (xlowast

Λ)]

(11)

Advances in Mathematical Physics 3

HereΔ(119909)

119895means the Laplacian in variable119909 isin xlowast(119895) Next ♯xlowast

stands for the cardinality of the particle configuration xlowast (ie♯xlowast = 119896 when xlowast isin 119872

(119896)) and the parameter 120581 is introducedin (17) (Symbol ♯ will be used for denoting the cardinalityof a general (finite) set for example ♯Λ means the numberof vertices in Λ) Further xlowast(119895119909)rarr (119895

1015840

119910)

Λdenotes the particle

configuration with the point 119909 isin xlowast(119895) removed and point 119910added to xlowast(1198951015840)

As in [1] we also consider a Hamiltonian HΛ|xlowastΓΛ

in anexternal field generated by a configuration xlowast

ΓΛ= xlowast(1198951015840)

119895

1015840isin Γ Λ isin 119872

lowastΓΛ where Γ sube Γ is a (finite or infinite)collection of vertices More precisely we only consider xlowast

ΓΛ

with ♯xlowast(1198951015840) le 120581 (see (17) below) and set

(HΛ|xlowastΓΛ

120601) (xlowastΛ)

=[

[

minus

1

2

sum

119895isinΛ

sum


(119909)

119895+ sum

119895isinΛ

sum

119909isinxlowast(119895)119880

(1)(119909)

+

1

2

sum

119895isinΛ

sum

1199091199091015840isinxlowast(119895)

1 (119909 = 119909

1015840)119880

(2)(119909 119909

1015840)

+

1

2

sum

1198951198951015840isinΛ

1 (119895 = 119895

1015840) 119869 (d (119895 119895

1015840))

times sum


119881 (119909 119909

1015840)]

]

120601 (xlowastΛ)

+ sum

119895isinΛ1198951015840isinΓΛ

119869 (d (119895 1198951015840))

times sum


119881 (119909 119909

1015840)120601 (xlowast

Λ)

+ sum

1198951198951015840isinΛ


times sum

119909isinxlowast(119895)

int

119872

V (d119910) [120601 (xlowast(119895119909)rarr (1198951015840

119910)


Λ)]

(12)

The novel elements in (11) and (12) compared with [1]are the presence of on-site potentials 119880(1) and 119880

(2) and thesummand involving transition rates 120582

1198951198951015840 ge 0 for jumps of a

particle from site 119895 to 1198951015840We will suppose that 120582

1198951198951015840 vanishes if the graph distance

d(119895 1198951015840) gt 1 We will also assume uniform boundedness

sup [1205821198951198951015840 (119909119872) 119895 119895

1015840isin Γ 119909 isin 119872] lt infin (13)

in view of (1) it implies that the total exit ratesum

1198951015840d(1198951198951015840)=1 120582119895119895

1015840(119909119872) from site 119895 is uniformly boundedThese conditions are not sharp and can be liberalized

The model under consideration can be considered as ageneralization of the Hubbard model [6] (in its bosonic ver-sion) Its mathematical justification includes the following(a) An opportunity to introduce a Fock space formalismincorporates a number of new features For instance afermonic version of the model (not considered here) emergesnaturally when the bosonic Fock space H(119894) is replaced bya fermonic one Another opening provided by this modelis a possibility to consider random potentials 119880(1) 119880(2) and119881 which would yield a sound generalization of the Mott-Andersonmodel (b) Introducing jumpsmakes a step towardsa treatment of a model of a quantum (Bose-) gas whereparticles ldquoliverdquo in a single Fock space For example a systemof interacting quantum particles is originally confined to aldquoboxrdquo in a Euclidean space with or without ldquointernalrdquo degreesof freedom In the thermodynamical limit the box expandsto the whole Euclidean space In a two-dimensional modelof a quantum gas one expects a phenomenon of invarianceunder space-translations one hopes to be able to address thisissue in future publications (c) A model with jumps can beanalysed by means of the theory of Markov processes whichprovides a developed methodology

Physically speaking the model with jumps covers a situ-ation where ldquolightrdquo quantum particles are subject to a ldquoran-domrdquo force and change their ldquolocationrdquo This class of modelsis interesting from the point of view of transport phenomenathat theymay display (An analogywith the famousAndersonmodel in its multiparticle version inevitably comes tomind cf eg [7]) Methodologically such systems occupyan ldquointermediaterdquo place between models where quantumparticles are ldquofixedrdquo forever to their designated locations (asin [1]) andmodels where quantumparticlesmove in the samespace (a Bose-gas considered in [8 9]) In particular thiswork provides a bridge between [1 8 9] reading this paperahead of [8 9] might help an interested reader to get through[8 9] at a much quicker pace

We would like to note an interesting problem of analysisof the small-mass limit (cf [10]) from the point of Mermin-Wagner phenomena

13 Assumptions on the Potentials The between-sites poten-tial 119881 is assumed to be of class 1198622 Consequently 119881 and itsfirst and second derivatives satisfy uniform bounds Namelyforall119909

1015840 119909

10158401015840isin 119872

minus119881(119909

1015840 119909

10158401015840)

10038161003816100381610038161003816

nablax119881(119909

1015840 119909

10158401015840)

10038161003816100381610038161003816

10038161003816100381610038161003816

nablaxx1015840119881(119909

1015840 119909

10158401015840)

10038161003816100381610038161003816

le 119881 (14)

Here 119909 and 1199091015840 run through the pairs of variables 119909 1199091015840 A simi-lar property is assumed for the on-site potential119880(1) (here weneed only a 1198621 smoothness)

minus119880

(1)(119909)

10038161003816100381610038161003816

nablax119880(1)

(119909)

10038161003816100381610038161003816

le 119880

(1)

119909 isin 119872 (15)

Note that for119881 and119880(1) the bounds are imposed on their neg-ative parts only

As to 119880(2) we suppose that (a)

119880

(2)(119909 119909

1015840) = +infin when 1003816

1003816100381610038161003816

119909 minus 119909

101584010038161003816100381610038161003816

le 120588 (16)


and (b) exist a 1198621-function (119909 119909

1015840) 997891rarr

119880

(2)(119909 119909

1015840) isin R such that

119880

(2)(119909 119909

1015840) =

119880

(2)(119909 119909

1015840) whenever 120588(119909 1199091015840) gt 120588 Here 120588(119909 1199091015840)

stands for the (flat) Riemannian distance between points119909 119909

1015840isin 119872 As a result of (16) there exists a ldquohard corerdquo of

diameter 120588 and a given atom cannot ldquoholdrdquo more than

120581 = lceil

V (119872)

V (119861 (120588))

rceil (17)

particles where V(119861(120588)) is the volume of a 119889-dimensional ballof diameter 120588 We will also use the bound

minus119880

(2)(119909 119909

1015840)

10038161003816100381610038161003816

nablax119880(2)

(119909 119909

1015840)

10038161003816100381610038161003816

le 119880

(2)

119909 119909

1015840isin 119872 (18)

Formally (16) means that the operators in (11) and (12) areconsidered for functions120601(xlowast

Λ) vanishingwhen in the particle

configuration xlowastΛ= xlowast(119895) 119895 isin Λ the cardinality ♯xlowast(119895) gt 120581

for some 119895 isin ΛThe function 119869 119903 isin (0infin) 997891rarr 119869(119903) ge 0 is assumed

monotonically nonincreasing with 119903 and obeying the relation119869(119897) rarr 0 as 119897 rarr infin where

119869 (119897) = sup[

[

sum

11989510158401015840isinΓ

119869 (d (1198951015840 119895

10158401015840)) 1 (d (1198951015840 11989510158401015840) ge 119897) 119895

1015840isin Γ

]

]

lt infin

(19)

Additionally let 119869(119903) be such that

119869

lowast= sup[

[

sum

1198951015840isinΓ

119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

119895 isin Γ]

]

lt infin (20)

Next we assume that the functions 119880(1) 119880(2) and 119881 are g-invariant forall119909 1199091015840 isin 119872 and g isin G

119880

(1)(119909) = 119880

(1)(g119909)

119880

(2)(119909 119909

1015840) = 119880

(2)(g119909 g119909

1015840)

119881 (119909 119909

1015840) = 119881 (g119909 g119909

1015840)

(21)

In the following we will need to bound the fugacity (oractivity cf (25)) 119911 in terms of the other parameters of themodel

119911119890

Θlt 1 where Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881) (22)

14 The Gibbs State in a Finite Volume Define the particlenumber operator N

Λ with the action

NΛ120601 (xlowast

Λ) = ♯xlowast

Λ120601 (xlowast

Λ) xlowast

Λisin 119872

lowastΛ (23)

Here for a given xlowastΛ= xlowast(119895) 119895 isin Λ ♯xlowast

Λstands for the total

number of particles in configuration xlowastΛ

♯xlowastΛ= sum

119895isinΛ

♯xlowast (119895) (24)

The standard canonical variable associated withNΛis activity

119911 isin (0infin)The Hamiltonians (11) and (12) are self-adjoint (on the

natural domains) in H(Λ) Moreover they are positivedefinite and have a discrete spectrum cf [14] Furthermoreforall119911 120573 gt 0 H

Λand H

Λ|xlowastΓΛ

give rise to positive-definite trace-class operators G

Λ= G

119911120573Λand G

Λ|xlowastΓΛ

= G119911120573Λ|xlowast

ΓΛ

GΛ= 119911

NΛ exp [minus120573H

Λ]

GΛ|xlowastΓΛ

= 119911


Λ|xlowastΓΛ

]

(25)

We would like to stress that the full range of variables 119911 120573 gt 0

is allowedhere because of the hard-core condition (16) it doesnot allow more than 120581♯Λ particles in Λ where ♯Λ stands forthe number of vertices in Λ However while passing to thethermodynamic limit we will need to control 119911 and 120573

Definition 1 We will call GΛand G

Λ|xlowastΓΛ

the Gibbs operatorsin volume Λ for given values of 119911 and 120573 (andmdashin the case ofG

Λ|xlowastΓΛ

mdashwith the boundary condition xlowastΓΛ

)The Gibbs operators in turn give rise to the Gibbs states120593Λ= 120593

120573119911Λand 120593

Λ|xΓΛ

= 120593120573119911Λ|x

ΓΛ

at temperature 120573minus1 andactivity 119911 in volume Λ These are linear positive normalizedfunctionals on the 119862

lowast-algebra BΛof bounded operators in

spaceHΛ

120593Λ(A) = trH

Λ

(RΛA)

120593Λ|xΓΛ

(A) = trHΛ

(RΛ|xΓΛ

A) A isin BΛ

(26)

where

RΛ=

GΛ

Ξ (Λ) with Ξ (Λ) = Ξ

119911120573(Λ) = trH

Λ

GΛ (27)

RΛ|xlowastΓΛ

=

GΛ|xΓΛ

Ξ (Λ | xlowastΓΛ

)

with Ξ (Λ | xlowastΓΛ

) = Ξ119911120573

(Λ | xlowastΓΛ

)

= trHΛ

(119911


Λ|xlowastΓΛ

])

(28)

The hard-core assumption (16) yields that the quantitiesΞ(Λ) and Ξ

119911120573(Λ | xlowast

ΓΛ) are finite formally these facts will

be verified by virtue of the Feynman-Kac representation

Definition 2 Whenever Λ0sub Λ the 119862lowast-algebraB

Λ0 is iden-

tified with the 119862lowast subalgebra inBΛformed by the operators

of the form A0otimes I

ΛΛ0 Consequently the restriction 120593Λ

0

Λof

state 120593Λto 119862lowast-algebraB

Λ0 is given by

120593Λ0

Λ(A

0) = trH

Λ0

(RΛ0

ΛA0) A

0isin B

Λ0 (29)

where

RΛ0

Λ= trH

ΛΛ0

RΛ (30)


Operators RΛ0

Λ(we again call them RDMs) are positive defi-

nite and have trHΛ0

RΛ0

Λ= 1 They also satisfy the compatibil-

ity property forallΛ0sub Λ

1sub Λ

RΛ0

Λ= trH

Λ1Λ0

RΛ1

Λ (31)

In a similar fashion one defines functionals 120593Λ0

Λ|xlowastΓΛ

and oper-

ators RΛ0

Λ|xlowastΓΛ

with the same properties

15 Limiting Gibbs States The concept of a limiting Gibbsstate is related to notion of a quasilocal 119862lowast-algebra see [14]For the class of systems under consideration the constructionof the quasilocal 119862lowast-algebraB

Γis done along the same lines

as in [1]BΓis the norm completion of the 119862lowast algebra (B0

Γ) =

lim ind119899rarrinfin

BΛ119899

Any family of positive-definite operatorsRΛ0

in spacesHΛ0 of trace one whereΛ0 runs through finite

subsets of Γ with the compatibility property

RΛ1

= trHΛ0Λ1

RΛ0

Λ

1sub Λ

0 (32)

determines a state ofBΓ see [12 13]

Finally we introduce unitary operators UΛ0(g) g isin G in

HΛ0

UΛ120601 (xlowast

Λ0) = 120601 (g

minus1xlowastΛ0) (33)

where

gminus1xlowast

Λ0 = g

minus1xlowast (119895) 119895 isin Λ

0

gminus1xlowast (119895) = g

minus1119909 119909 isin xlowast (119895)

(34)

Theorem3 Assuming the conditions listed above for all 119911 120573 isin

(0 +infin) satisfying (22) and a finite Λ

0sub Γ operators RΛ

0

Λ

form a compact sequence in the trace-norm topology in HΛ0

as Λ Γ Furthermore given any family of (finite or infinite)sets Γ = Γ(Λ) sube Γ and configurations xlowast

ΓΛ= xlowast(119894) 119894 isin Γ Λ

with ♯xlowast(119894) lt 120581 operatorsRΛ0

Λ|xΓΛ

also forma compact sequence

in the trace-norm topology Any limit point RΛ0

for RΛ0

Λ or

RΛ0

Λ|xlowastΓΛ

as Λ Γ is a positive-definite operator in H(Λ

0)

of trace one Moreover if Λ1sub Λ

0 and RΛ0

and RΛ1

are thelimits forRΛ

0

ΛandRΛ

1

Λor forRΛ

0

Λ|xlowastΓΛ

andRΛ1

Λ|xlowastΓΛ

along the samesubsequence Λ

119904 Γ then the property (32) holds true

Consequently theGibbs states120593Λand120593

Λ|xlowastΓΛ

form compactsequences as Λ Γ

Remark 4 In fact the assertion of Theorem 3 holds withoutassuming the bidimensionality condition on graph (ΓE)only under an assumption that the degree of the vertices inΓ is uniformy bounded

Definition 5 Any limit point 120593 for states 120593Λand 120593

Λ|xlowastΓΛ

iscalled a limiting Gibbs state (for given 119911 120573 isin (0 +infin))

Theorem 6 Under the condition (22) any limiting pointRΛ0

for RΛ

0

Λ or RΛ

0

Λ|xlowastΓΛ

as Λ Γ is a positive-definite operatorof trace one commuting with operators U

Λ0(g) forallg isin G

UΛ0(g)

minus1RΛ0

UΛ0 (g) = RΛ

0

(35)

Accordingly any limiting Gibbs state 120593 of B determined by afamily of limiting operators RΛ

0

obeying (35) satisfies the cor-responding invariance property forall finite Λ0

sub Γ any A isin BΛ0

and g isin G

120593 (A) = 120593 (UΛ0(g)

minus1AUΛ0 (g)) (36)

Remarks (1) Condition (22) does not imply the uniqueness ofan infinite-volume Gibbs state (ie absence of phase transi-tions)

(2) Properties (35) and (36) are trivially fulfilled for thelimiting points RΛ

0

and 120593 of families RΛ0

Λ and 120593

Λ How-

ever they require a proof for the limit points of the familiesRΛ0

Λ|xlowastΓΛ

and 120593Λ|xlowastΓΛ

The set of limiting Gibbs states (which is nonempty due

to Theorem 3) is denoted by G0 In Section 3 we describe aclassG sup G0 of states of119862lowast-algebraB satisfying the FK-DLRequation similar to that in [1]

2 Feynman-Kac Representations forthe RDM Kernels in a Finite Volume

21 The Representation for the Kernels of the Gibbs OperatorsA starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernels K

Λ(xlowast

Λ ylowast

Λ) =

K120573119911Λ

(xlowastΛ ylowast

Λ) and F

Λ(xlowast

Λ ylowast

Λ) = F

120573119911Λ(xlowast

Λ ylowast

Λ) of operators

GΛand R

Λ

Definition 7 Given (119909 119894) (119910 119895) isin 119872 times Γ 119882120573

(119909119894)(119910119895)denotes

the space of path or trajectories 120596 = 120596(119909119894)(119910119895)

in 119872 times Γ oftime-length 120573 with the end-points (119909 119894) and (119910 119895) Formally120596 isin 119882

120573

(119909119894)(119910119895)is defined as follows

120596 120591 isin [0 120573] 997891997888rarr 120596 (120591) = (119906 (120596 120591) 119897 (120596 120591)) isin 119872 times Γ

120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)

120596 has finitely many jumps on [0 120573]

if a jump occurs at time 120591 then d [119897 (120596 120591minus) 119897 (120596 120591)]=1

(37)

The notation 120596(120591) and its alternative (119906(120596 120591) 119897(120596 120591)) forthe position and the index of trajectory 120596 at time 120591 will beemployed as equal in rights We use the term the temporalsection (or simply the section) of path 120596 at time 120591

Definition 8 Let xlowastΛ

= xlowast(119894) 119894 isin Λ isin 119872

lowastΛ and ylowastΛ

=

ylowast(119895) 119895 isin Λ isin 119872

lowastΛ be particle configurations overΛ with♯xlowast

Λ= ♯ylowast

Λ A matching (or pairing) 120574 between xlowast

Λand ylowast

Λis


defined as a collection of pairs [(119909 119894) (119910 119895)]120574 with 119894 119895 isin Λ

119909 isin xlowast(119894) and 119910 isin ylowast(119895) with the properties that (i) forall119894 isin Λ

and 119909 isin xlowast(119894) there exist unique 119895 isin Λ and 119910 isin ylowast(119895)such that (119909 119894) and (119910 119895) form a pair and (ii) forall119895 isin Λ and119910 isin ylowast(119895) there exist unique 119894 isin Λ and 119909 isin xlowast(119894) suchthat (119909 119894) and (119910 119895) form a pair (Owing to the condition♯xlowast

Λ= ♯ylowast

Λ these properties are equivalent) It is convenient

to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]

Next consider the Cartesian product

119882

120573

xlowastΛylowastΛ120574= times

119894isinΛ

times

119909isinxlowast(119894)119882

120573

(119909119894)120574(119909119894) (38)

and the disjoint union

119882

120573

xlowastΛylowastΛ

= ⋃

120574

119882

120573

xlowastΛylowastΛ120574 (39)

Accordingly an element 120596Λ

isin 119882

120573

xlowastΛylowastΛ120574in (38) represents a

collection of paths 120596119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 120573

starting at (119909 119894) and ending up at 120574(119909 119894) We say that 120596Λis a

path configuration in (or over) Λ

The presence of matchings in the above construction is afeature of the bosonic nature of the systems under consider-ation

We will work with standard sigma algebras (generated bycylinder sets) in119882

120573

(119909119894)(119910119895)119882120573

xlowastΛylowastΛ120574 and119882

120573

xlowastΛylowastΛ

Definition 9 In what follows 120585(120591) 120591 ge 0 stands for theMarkov process on 119872 times Γ with cadlag trajectories deter-mined by the generator L acting on a function (119909 119894) isin 119872 times

Λ 997891rarr 120595(119909 119894) by

L120595 (119909 119894) = minus

1

2

Δ120595 (119909 119894)

+ sum

119895d(119894119895)=1

120582119894119895int

119872

V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]

(40)

In the probabilistic literature such processes are referred toas Levy processes see for example [14]

Pictorially a trajectory of process 120585 moves along 119872

according to the Brownian motion with the generator minusΔ2and changes the index 119894 isin Γ from time to time in accor-dance with jumps occurring in a Poisson process of ratesum

119895d(119894119895)=1 120582119894119895 In other words while following a Brownian

motion rule on 119872 having index 119894 isin Γ and being at point119909 isin 119872 themoving particle experiences an urge to jump from119894 to a neighboring vertex 119895 and to a point 119910 at rate 120582

119894119895V(d119910)

After a jump the particle continues the Brownian motion on119872 from 119910 and keeps its new index 119895 until the next jump andso on

For a given pairs (119909 119894) (119910 119895) isin 119872 times Γ we denote byP120573

(119909119894)(119910119895)the nonnormalised measure on 119882

120573

(119909119894)(119910119895)induced

by 120585 That is under measure P120573

(119909119894)(119910119895)the trajectory at time

120591 = 0 starts from the point 119909 and has the initial index 119894 whileat time 120591 = 120573 it is at the point 119910 and has the index 119895The value119901(119909119894)(119910119895)

= P120573

(119909119894)(119910119895)(119882

120573

(119909119894)(119910119895)) is given by

119901(119909119894)(119910119895)

= 1 (119894 = 119895) 119901

120573

119872(119909 119910) exp[

[

minus120573 sum

119895d(119894119895)=1

120582119894119895]

]

+ sum

119896ge1

sum

1198970=1198941198971119897119896119897119896+1

=119895

prod

0le119904le119896

1 (d (119897119904 119897119904+1

) = 1)

times 120582119897119904119897119904+1

int

120573

0

d120591119904exp[

[

minus (120591119904+1

minus 120591119904) sum

119895d(119897119904119895)=1

120582119897119904119895]

]

times 1 (0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119896lt 120591

119896+1= 120573)

(41)

where 119901120573

119872(119909 119910) denotes the transition probability density for

the Brownian motion to pass from 119909 to 119910 on119872 in time 120573

119901

120573

119872(119909 119910) =

1

(2120587120573)

1198892

times sum

119899=(1198991119899119889)isinZ119889

exp(minus

1003816100381610038161003816

119909 minus 119910 + 119899

1003816100381610038161003816

2

2120573

)

(42)

In view of (13) the quantity 119901(119909119894)(119910119895)

and its derivatives areuniformly bounded

119901(119909119894)(119910119895)

10038161003816100381610038161003816

nabla119909119901(119909119894)(119910119895)

10038161003816100381610038161003816

10038161003816100381610038161003816

nabla119910119901(119909119894)(119910119895)

10038161003816100381610038161003816

le 119901119872

119909 119910 isin 119872 119894 119895 isin Γ

(43)

where 119901119872

= 119901119872(120573) isin (0 +infin) is a constant

We suggest a term ldquonon-normalised Brownian bridgewith jumpsrdquo for themeasure but expect that a better termwillbe proposed in future

Definition 10 Suppose that xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872

lowastΛ andylowastΛ= ylowast(119895) 119895 isin Λ isin 119872

lowastΛ are particle configurations overΛ with ♯xlowast

Λ= ♯ylowast

Λ Let 120574 be a pairing between xlowast

Λand ylowast

Λ

Then Plowast

xlowastΛylowastΛ120574denotes the product measure on119882

120573

xlowastΛylowastΛ120574

P120573


119894isinΛ

times

119909isinxlowast(119894)P120573

(119909119894)120574(119909119894) (44)

Furthermore P120573

xlowastΛylowastΛ

stands for the sum measure on119882

120573

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

= sum

120574

P120573


According to Definition 10 under the measure P120573


the trajectories 120596119909119894

isin 119882

120573

(119909119894)120574(119909119894)constituting 120596

Λare inde-

pendent components (Here the term independence is usedin the measure-theoretical sense)


As in [1] we will work with functionals on119882

120573

xlowastΛylowastΛ120574repre-

senting integrals along trajectories The first such functionalhΛ(120596

Λ) is given by

hΛ (120596Λ) = sum

119894isinΛ

sum

119909isinxlowast(119894)h119909119894 (120596

119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum

119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909

1015840

1198941015840

)

times (120596119909119894 120596

11990910158401198941015840)

(46)

Here introducing the notation 119906119909119894(120591) = 119906(120596

119909119894 120591) and

11990611990910158401198941015840(120591) = 119906(120596

11990910158401198941015840 120591) for the positions in 119872 of paths 120596

119909119894isin

119882

120573

(119909119894)120574(119909119894)and 120596

11990910158401198941015840 isin 119882

120573

(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define

h119909119894 (120596119909119894) = int

120573

0

d120591119880(1)(119906

119894119909(120591)) (47)

Next with 119897119909119894(120591) and 119897

11990910158401198941015840(120591) standing for the indices of 120596

119909119894

and 12059611990910158401198941015840 at time 120591

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 120596

11990910158401198941015840)

= int

120573

0

d120591[

[

sum

1198951015840isinΓ

119880

(2)(119906

119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119897

11990910158401198941015840 (120591))

+

1

2

sum

(119895101584011989510158401015840)isinΓtimesΓ

119869 (d (1198951015840 119895

10158401015840))

times 119881 (119906119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119895

10158401015840= 119897

11990910158401198941015840 (120591))

]

]

(48)

Next consider the functional hΛ(120596Λ| xlowast

ΓΛ) for xlowast

ΓΛ=

xlowast(119895) 119895 isin Γ Λ As before we assume that ♯xlowast(119895) le 120581Define

hΛ (120596Λ| xlowast

ΓΛ) = hΛ (120596

Λ) + hΛ (120596

Λ|| xlowast

ΓΛ) (49)

Here hΛ(120596Λ) is as in (46) and

hΛ (120596Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


1015840

1198941015840

)

times (120596119909119894 (119909

1015840 119894

1015840))

(50)

where in turn

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 (119909

1015840 119894

1015840))

= int

120573

0

d120591 [119880(2)(119906

119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119894

1015840)

+ sum

119895isinΓ119895 = 1198941015840

119869 (d (119895 1198941015840))

times119881 (119906119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119895)]

(51)

The functionals hΛ(120596Λ) and hΛ(120596

Λ|| xlowast

ΓΛ) are interpreted as

energies of path configurations Compare (214) and (238)

in [1]Finally we introduce the indicator functional 120572

Λ(120596

Λ)

120572Λ(120596

Λ) =

1 if index 119897119909119894(120591) isin Λ

forall120591 isin [0 120573] 119894 isin Λ 119909 isin xlowast (119894) 0 otherwise

(52)

It can be derived from known results [11 15ndash17] (for a directargument see [18]) that the following assertion holds true

Lemma 11 For all 119911 120573 gt 0 and a finite Λ the Gibbs operatorsG

Λand G

Λ|xlowastΓΛ

act as integral operators inH(Λ)

(GΛ120601) (xlowast

Λ) = int

119872lowastΛ

prod

119895isinΛ

prod

119910isinylowast(119895)

V (d119910)KΛ(xlowast

Λ ylowast

Λ)120601 (ylowast

Λ)

(GΛ|xlowastΓΛ

120601) (xlowastΛ)

= int

119872lowastΛ

prod

119895isinΛ

prod

119910isinylowast(119895)V (d119910)K

Λ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ)120601 (ylowast

Λ)

(53)

Moreover the integral kernels KΛ(xlowast

Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ|

xlowastΓΛ

) vanish if ♯xlowastΛ

= ♯ylowastΛ On the other hand when ♯xlowast

Λ=

♯ylowastΛ the kernels K

Λ(xlowast

Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ) admit the

following representations

KΛ(xlowast

Λ ylowast

Λ) = 119911

♯xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)

times exp [minushΛ (120596Λ)]

(54)

KΛ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ)

= 119911

♯ xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)

times exp [minushΛ (120596Λ| xlowast

ΓΛ)]

(55)

The ingredients of these representations are determined in (46)ndash(51)


Remark 12 As before we stress that owing to (16) and (17)a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 120596

Λ= 120596

119909119894 such that

forall120591 isin [0 120573] and forall119895 isin Γ the number of paths 120596119909119894

with index119897119909119894(120591) = 119895 is less than or equal to 120581 Likewise the integral in

the RHS of (55) receives a non-zero contribution only fromconfigurations120596

Λ= 120596

lowast

119909119894 such that forall site 119895 isin Γ the number

of paths 120596119909119894

with index 119897119909119894(120591) = 119895 plus the cardinality ♯xlowast(119895)

does not exceed 120581

22 The Representation for the Partition Function The FKrepresentations of the partition functions Ξ(Λ) = Ξ

120573119911(Λ)

in (27) and Ξ(Λ | xlowastΓΛ

) in (146) reflect a specific characterof the traces trG

Λand trG

Λ|xlowastΓΛ

in H(Λ) The source of acomplication here is the jump terms in the Hamiltonians119867

Λ

and119867Λ|xlowastΓΛ

in (11) and (12) respectively In particular we willhave to pass from trajectories of fixed time-length 120573 to loopsof a variable time length To this end a given matching 120574

is decomposed into a product of cycles and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-lengthmultiple of120573 (A similar constructionhas been performed in [18])

To simplify the notation we omit wherever possible theindex 120573

Definition 13 For given (119909 119894) (119910 119895) isin 119872 times Γ the symbol119882

lowast

(119909119894)(119910119895)denotes the disjoint union

119882

lowast

(119909119894)(119910119895)= ⋃

119896=01

119882

119896120573

(119909119894)(119910119895) (56)

In other words119882lowast

(119909119894)(119910119895)is the space of pathsΩlowast

= Ω

lowast

(119909119894)(119910119895)

in119872timesΓ of a variable time-length 119896120573 where 119896 = 119896(Ω

lowast

) takesvalues 1 2 and called the lengthmultiplicity with the end-points (119909 119894) and (119910 119895)The formal definition follows the sameline as in (37) and we again use the notation Ω

lowast

(120591) and thenotation (119906(Ω

lowast

120591) 119897(Ω

lowast

120591)) for the pair of the position andthe index of path Ω

lowast at time 120591 Next we call the particleconfiguration Ω

lowast

(120591 + 120573119898) 0 le 119898 lt 119896(Ω

lowast

) the temporalsection (or simply the section) of Ωlowast at time 120591 isin [0 120573] Wealso callΩlowast

(119909119894)(119910119895)isin 119882

lowast

(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))

A particular role will be played by closed paths (loops)with coinciding endpoints (where (119909 119894) = (119910 119895)) Accord-ingly we denote by 119882

lowast

119909119894the set 119882lowast

(119909119894)(119909119894) An element of

119882

lowast

119909119894is denoted by Ω

lowast

119909119894or in short by Ω

lowast and called a loopat vertex 119894 (The upper index lowast indicates that the lengthmultiplicity is unrestricted) The length multiplicity of a loopΩ

lowast

119909119894isin 119882

lowast

119909119894is denoted by 119896(Ωlowast

119909119894) or 119896

119909119894 It is instructive to

note that as topological object a given loop Ωlowast admits amultiple choice of the initial pair (119909 119894) it can be representedby any pair (119906(Ωlowast

120591) 119897(Ω

lowast 120591)) at a time 120591 = 119897120573 where 119897 =

1 119896(Ωlowast) As above we use the term the temporal sectionat time 120591 isin [0 120573] for the particle configuration Ω

lowast

119909119894(120591 +

120573119898) 0 le 119898 lt 119896119909119894 and employ the alternative notation

(119906(120591 + 120573119898Ω

lowast) 119897(120591 + 120573119898Ω

lowast)) addressing the position and

the index ofΩlowast at time 120591 + 120573119898 isin [0 120573119896(Ω

lowast)]

Definition 14 Suppose xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872

lowastΛ and ylowastΛ=


lowastΛ are particle configurations over Λwith ♯xlowast

Λ= ♯ylowast

Λ Let 120574 be a matching between xlowast

Λand ylowast

Λ We

consider the Cartesian product

119882

lowast


119894isinΛ

times

119909isinxlowast(119894)119882

lowast

(119909119894)120574(119909119894) (57)


119882

lowast

xlowastΛylowastΛ

= ⋃

120574

119882

lowast


Accordingly an element ΩlowastΛisin 119882

lowast


collection of paths Ωlowast119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 119896120573

starting at (119909 119894) and ending up at (119910 119895) = 120574(119909 119894) We say thatΩ

lowast

Λisin 119882

lowast

xlowastΛylowastΛ

is a path configuration in (or over) ΛAgain loops play a special role and deserve a particular

notation Namely119882lowast

xlowastΛ

denotes the Cartesian product

119882

lowast

xlowastΛ

= times

119894isinΛ

times

119909isinxlowast(119894)119882

lowast

119909119894 (59)

and 119882

lowast

Λstands for the disjoint union (or equivalently the

Cartesian power)

119882

lowast

Λ= ⋃

xlowastΛisin119872lowastΛ

119882

lowast

xlowastΛ

= times

119894isinΛ

119882

lowast

119894

where 119882

lowast

119894= ⋃

xlowastisin119872lowast( times

119909isinxlowast119882

lowast

119909119894)

(60)

Denote by Ωlowast = Ωlowast(119894) 119894 isin Λ isin 119882

lowast

Λa collection of

loop configurations at vertices 119894 isin Λ starting and ending upat particle configurations xlowast(119894) isin 119872

lowast (note that some of theΩlowast(119894)rsquos may be empty) The temporal section (or in shortthe section) Ωlowast(120591) of Ωlowast at time 120591 is defined as the particleconfiguration formed by the pointsΩlowast

119909119894(120591+120573119898)where 119894 isin Λ

119909 isin xlowast(119894) and 0 le 119898 lt 119896119909119894

As before consider the standard sigma algebras of subsetsin the spaces 119882lowast

(119909119894)(119910119895) 119882

119909119894 119882lowast

xlowastΛylowastΛ120574 119882lowast

xlowastΛylowastΛ

119882lowast

xlowastΛ

and 119882

lowast

Λ

introduced in Definitions 13 and 14 In particular the sigmaalgebra of subsets in119882lowast

Λwill be denoted by W

Λ we comment

on some of its specific properties in Section 31 (An infinite-volume version119882

lowast

Γof119882lowast

Λis treated in Section 32 and after)

Definition 15 Given points (119909 119894) (119910 119895) isin 119872 times Γ we denoteby Plowast

(119909119894)(119910119895)the sum measure on119882

lowast

(119909119894)(119910119895)

Plowast

(119909119894)(119910119895)= sum

119896=01

P119896120573

(119909119894)(119910119895) (61)

Further Plowast

119909119894denotes the similar measure on119882

lowast

119909119894

Plowast

119909119894= sum

119896=01

P119896120573

119909119894 (62)





=



Λ= ♯ylowast


Λand ylowast

Λ and we

define the product measure Plowast


Plowast


119894isinΛ

times

119909isinxlowast(119894)Plowast

(119909119894)120574(119909119894) (63)

and the sum measure

Plowast

xlowastΛylowastΛ

= sum

120574

Plowast


Next symbolPlowast

xlowastΛ

stands for the product measure on119882lowast

xlowastΛ

Plowast

xlowastΛ

= times

119894isinΛ

times


119909119894 (65)

Finally dΩlowastΛyields the measure on119882

lowast

Λ

dΩlowastΛ= dxlowast

Λtimes P

lowast

xlowastΛ

(dΩlowastΛ) (66)

Here for xlowastΛ

= xlowast(119894) 119894 isin Λ we set dxlowastΛ

=

prod119894isinΛ

prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-

sponding factors are trivial measures sitting on the emptyconfigurations

We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast

Λ) and hΛ(Ωlowast

Λ| xlowast

ΓΛ) which are

modifications of the above functionals hΛ(120596Λ) and hΛ(120596

Λ|

xlowastΓΛ

) confer (46) and (50) Say for a loop configurationΩlowastΛ=

Ω

lowast

119909119894 over Λ with an initial and final particle configuration

xlowastΛ= xlowast(119894) 119894 isin Λ

hΛ (Ωlowast

Λ)

= sum

119894isinΛ

sum

119909isinxlowast(119894)h119909119894 (Ωlowast

119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


times 1 ((119909 119894) = (119909

1015840 119894

1015840)) h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

x119894 Ωlowast

11990910158401198941015840)

(67)

To determine the functionals h119909119894(Ωlowast

119909119894) and h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

119909119894

Ω

lowast

11990910158401198941015840) we set for given 119898 = 0 1 119896

119909119894minus 1 and 119898

1015840=

0 1 11989611990910158401198941015840 minus 1

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

119909119894)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

119909119894)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(68)

A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the

index 119897119909119894(120591 + 120573119898Ω

lowast

119909119894) and 119906

119894119909(120591 + 120573119898) for the position

119906(120591 + 120573119898Ω

lowast

119909119894) for Ωlowast

119909119894(120591) isin 119872 times Γ of the section Ω

lowast

119909119894(120591)

of the loop Ω

lowast

119909119894at time 120591 and similarly with 119897

11989410158401199091015840(120591 + 120573119898

1015840)

and 11990611989410158401199091015840(120591 + 120573119898

1015840) (Note that the pairs (119909 119894) and (119909

1015840 119894

1015840)may

coincide) Then

h(119909119894) (Ωlowast

119909119894)

= int

120573

0

d120591[

[

sum

0le119898lt119896119909119894

119880

(1)(119906

119909119894(120591 + 120573119898))

+ sum

0le119898lt1198981015840lt119896119909119894

sum

119895isinΓ

1 (119897119909119894(120591 + 120573119898)

= 119895 = 119897119909119894(120591 + 120573119898

1015840))

times119880

(2)(119906

119909119894(120591 + 120573119898) 119906

119909119894(120591 + 120573119898

1015840))

]

]

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840 )

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(69)

Next the functional119861(ΩlowastΛ) takes into account the bosonic

character of the model

119861 (Ωlowast

Λ) = prod

119894isinΛ

prod


119911

119896119909119894

119896119909119894

(70)

The factor 119896minus1119909119894

in (70) reflects the fact that the starting pointof a loop Ω

lowast

119909119894may be selected among points 119906(120573119898Ω

lowast

119909119894)

arbitrarily


Next we define the functional hΛ(ΩlowastΛ| xlowast

ΓΛ) for xlowast

ΓΛ=

xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set

hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)

Here hΛ(ΩlowastΛ) is as in (67) and

hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)

As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast

Λ|| xlowast

ΓΛ) have a

natural interpretation as energies of loop configurationsFinally as before the functional 120572

Λ(Ωlowast

Λ) is the indicator

that the collection of loopsΩlowastΛ= Ω

lowast

119909119894 does not quit Λ

120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ

forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894

0 otherwise(74)

Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])

Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast

ΓΛ) in (27)

and (28) admit the representations as converging integrals

Ξ (Λ) = int

119882lowast

Λ

dΩlowastΛ119861 (Ω

lowast

Λ) 120572

Λ(Ω

lowast

Λ) exp [minushΛ (Ω

lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)

with the ingredients introduced in (60)ndash(74)

Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast

Λ= Ω

lowast

119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and

120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897

119909119894(120591+119898120573))

with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581

Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)

23 The Representation for the RDM Kernels Let Λ0 Λ be

finite sets Λ0sub Λ sub Γ The construction developed in

Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ

0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ

In accordancewith Lemma 11 and the definition ofRΛ0

Λin (31)

the operator RΛ0

Λacts as an integral operator inH(Λ

0)

(RΛ0

Λ120601) (xlowast0)

= int

119872lowastΛ0

prod

119895isinΛ0

prod

119910isinylowast(119895)V (d119910) FΛ

0

Λ(xlowast0 ylowast0)120601 (ylowast0)

(77)

where

FΛ0

Λ(xlowast0 ylowast0)

= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ(xlowast0 or zlowast

ΛΛ0 ylowast0 or zlowast

ΛΛ0)

=

ΞΛ0

Λ(xlowast0 ylowast0 Λ Λ

0)

Ξ (Λ)

(78)

We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast

Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =

ylowast(119895) 119895 isin Λ

0 overΛ0 Next xlowast0orzlowast

ΛΛ0 ylowast0orzlowast

ΛΛ0 denotes

the concatenated configurations over ΛSimilarly the RDM RΛ

0

Λ|xlowastΓΛ

is determined by its integral

kernel FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0) again admitting the representation

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0) =ΞΛ0


0| xlowast

ΓΛ)

Ξ (Λ)

(79)

As in [1] we call FΛ0

Λand FΛ

0

Λ|xlowastΓΛ

the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0


0) and Ξ

Λ0


0| xlowast

ΓΛ) in

(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions


Definition 19 Repeating (57)-(58) symbol 119882lowast

xlowast0 ylowast0 denotesthe disjoint union ⋃

1205740119882

lowast

xlowast0 ylowast01205740 over matchings 120574

0

between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882

lowast

xlowast0 ylowast01205740 yields a collection of pathsΩlowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)

lying in 119872 times Γ Each path Ω

lowast

(119909119894)1205740(119909119894)

has time lengths120573119896

(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574

0(119909 119894)

where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0

the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast

xlowast0 ylowast01205740 on 119882

lowast

xlowast0 ylowast01205740 and Plowast

xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0

The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH

ΛΛ0

in HΛ The meaning of new ingredients

in (79)ndash(82) is explained below

Lemma 20 The quantity ΞΛ0


0) emerging in

(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0

ΞΛ0


0)

= int

119882lowast

xlowast0ylowast0Plowast

xlowast0 ylowast0 (dΩlowast0

) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0

) 1 (Ωlowast0 isin FΛ0

)

times exp [minushΛ0

(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0

dΩlowastΛΛ0119861 (Ω

lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)

times exp [minushΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)

Similarly the quantity ΞΛ0


0| xlowast

ΓΛ) from (79)

vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0

ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)

times 1 (ΩlowastΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)

These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ

Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω

lowast0

) 1(Ωlowast0 isin

FΛ0

) 1(ΩlowastΛΛ0 isin FΛ

0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (79)ndash(83)(The functionals 119861(Ωlowast

ΛΛ0) and 120572

Λ(Ωlowast

ΛΛ0) are defined as (70)

and (74) respectively replacing Λ with Λ Λ

0)To this end let Ωlowast0 = Ω

lowast

xlowast0ylowast01205740 isin 119882

lowast

xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ

lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast

119909119894in short) with end points (119909 119894)

and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)

The functional 120572Λ(Ω

lowast0

) is again an indicator

120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)

Now let us define the indicator function 1(sdot isin FΛ0

) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ

0

) equals one if and onlyif every path Ω

lowast

(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896

(119909119894)(119910119895)

starting at (119909 119894) isin 119872timesΛ

0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ

0 remains in119872times (Λ Λ

0) at the intermediate times 120573119897

for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)

= 1 this is not a restriction)Furthermore suppose that Ωlowast

ΛΛ0 = Ωlowastxlowast

ΛΛ0

is a loop

configuration over Λ Λ

0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ

0 represented by a collection of

loops Ωlowast

119909119894 119894 isin Λ Λ

0 119909 isin xlowast(119894) Then 1(Ωlowast

ΛΛ0 isin FΛ

0

) = 1

if and only if each loop Ω

lowast

119909119894of time-length 120573119896

119909119894 beginning


and finishing at (119909 119894) isin 119872 times (Λ Λ

0) does not enter the set

119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894

= 1 this is not a restriction)The functional hΛ

0

(Ωlowast0

) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

) in(81) represents the energy of the loop configurationΩlowast

ΛΛ0 in

the potential field generated by the path configurationΩlowast0

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast

ΛΛ0) + h (Ωlowast0 || Ωlowast

ΛΛ0)

(88)

Here the summand hΛΛ0

(ΩlowastΛΛ0) yields the energy of the

loop configurationΩlowastΛΛ0 again confer (67) Further the term

h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ

lowast0

and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω

lowast

119909119894 isin

119882

lowast

xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω

lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set

h (Ωlowast0 || ΩlowastΛΛ0)

= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)

Here for a pathΩlowast

119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)

Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast

(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898

1015840) of Ωlowast

(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898

1015840 respectively

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)

Further the functional hΛ0

(Ωlowast0

| xlowastΓΛ

) in (82) isdetermined as in (71)ndash(73) with Ω

lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast

ΛΛ0) + h (Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)

Here again the summand hΛΛ0

(ΩlowastΛΛ0) is determined as in

(67) Next the term hΛΛ0

(ΩlowastΛΛ0 || Ω

lowast0

or xlowastΓΛ

) is definedsimilarly to (72)-(73)

hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)

have a natural interpretation as energies of loop configura-tions

Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω

lowast0

ΩlowastΛΛ0)

such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ

0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)

incident to the loops of the configuration ΩlowastΛΛ0 does not

exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast

ΛΛ0) such that the above

inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ

0

Λ(Λ Λ

0| Ω

lowast0

) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ

0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ

Λ0

Λ(ΛΛ

0| Ω

lowast0

) can jumpwithin volumeΛ only (owing to the indicator functional 120572

Λ)

On the other hand the presence of the indicator functional1(Ωlowast

ΛΛ0 isin FΛ

0

) in the integral (reflected in the upperscript

Λ

0 and the roof sign in the notation ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ

0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

) in (83) itis a particular example of a partition function in the volumeΛ Λ

0 with a boundary conditionΩlowast0 or xlowastΓΛ

Other useful types of partition functions are Ξ

Γ0(Λ |

Ωlowast0

or ΩlowastΓ1) and ΞΓ0(Λ | Ω

lowast0

or ΩlowastΓ1 or xlowast

Γ2) where the sets of

vertices Λ Λ0 Γ0 Γ1 and Γ

2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ

0lt +infin Accordingly Ωlowast0 is a (finite) configu-

ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration

over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration

over Γ

2 The partition functions ΞΓ0(Λ | Ω

lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0

orΩlowastΓ1 or xlowast

Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ

dΩlowastΛ120572Γ0 (Ω

lowast

Λ) 119861 (Ω

lowast

Λ)

times exp [minushΛ (Ωlowast

Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)

with the indicator 120572Γ0 as in (74) These partition functions

feature loop configurations Ωlowastformed by loops Ωlowast

119909119894 119894 isin

Λwhich start and finish in

Λ are confined to Γ

0 and movein a potential field generated by Ωlowast0 or Ωlowast

Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast

119909119894 119909 isin xlowast(119894) 119894 isin Γ

1 orΩlowast

Γ1 or xlowast

Γ2

where xlowastΓ2 = xlowast(119894) 119894 isin Γ

2 (The latter can be understood

as the concatenation of the loop configuration ΩlowastΓ1 over Γ1

and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ

2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number

♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)

does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)

Such ldquomodifiedrdquo partition functions will be used in forth-coming sections

24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs

states 120593Λand 120593

Λ|xlowastΓΛ

give rise to probability measures 120583Λand

120583Λ|xlowastΓΛ

on the sigma algebraWΛof subsets of119882lowast

Λ The sigma

algebra WΛis constructed by following the structure of the

space 119882

lowast

Λ(a disjoint union of Cartesian products) confer

Definition 16 The measures 120583Λand 120583

Λ|xlowastΓΛ

are determined


by their Radon-Nykodym derivative 119901Λand 119901

Λ|xlowastΓΛ

relative tothe measure dΩlowast

Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ

0sub Λ the sigma algebra W

Λ0 is naturally identified

with a sigma subalgebra of WΛ The restrictions of 120583

Λ

to WΛ0 and 120583

Λ|xlowastΓΛ

are denoted by 120583Λ0

Λand 120583Λ

0

Λ|xlowastΓΛ

thesemeasures are determined by their Radon-Nikodym deriva-tives 119901

Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ

(dΩlowastΛ0)dΩlowast

Λ0

The first key property of the measures 120583Λand 120583

Λ|xlowastΓΛ

isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ

Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ

0sub Λ

1015840sub Λ

the probability density 119901Λ0

Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) = exp [minushΛ

0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)

and the conditional partition functions ΞΛ(Λ

1015840 Λ

0| Ωlowast or

ΩlowastΛΛ1015840) and ΞΛ(Λ1015840

| ΩlowastΛΛ1015840) are determined as in (96)

Similarly for 119901Λ0

Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840 or xlowast

ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ

(dΩlowastΛΛ1015840)

(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or

ΩlowastΛΛ1015840 or xlowast

ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast


ΓΛ) are determined as in

(96)

As in [1] (100) and (102) mean that the conditional den-sities 119901Λ

0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0 | Ωlowast

ΛΛ1015840) relative

to 120590-algebra WΛΛ1015840

coincide respectively with 119902

Λ0

ΛΛ1015840(Ω

lowast0|

ΩlowastΛΛ1015840) and 119902

Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast


ΓΛ) for 120583ΛΛ

1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ

mdashaaΩlowastΛΛ1015840 isin 119882

lowast

ΛΛ1015840 and aaΩlowast

Λ0 isin 119882

lowast

Λ0

As in [1] we call the expressions 119902Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast


ΓΛ) as well as the expressions

119902

Λ0

ΛΛ1015840(Ω

lowast0

| ΩlowastΛΛ1015840) and 119902

Λ0

ΛΛ1015840(Ω

lowast0

| ΩlowastΛΛ1015840 or xlowast

ΓΛ)

appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ

0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) from (110)-(111) and the quantity

119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) from (112)-(113)The second property is that the RDMKs FΛ

0

Λ(xlowast0 ylowast0) and

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0) are related to the measures 120583Λand 120583

Λ|xlowastΓΛ1015840

Again the proof of this fact is done by inspection

Lemma 22 The RDMK FΛ0

Λ(xlowast0 ylowast0) is expressed as follows

forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)

times 119861 (Ωlowast0


)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

(dΩlowastΛΛ1015840) 1 (Ωlowast

ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0

orΩlowastΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0

orΩlowastΛΛ1015840 or xlowast

ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)

Here the partition functions ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0

or ΩlowastΛΛ1015840) and

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ) are determined in (81) and

(83)

Remark 23 Summarizing the above observations the mea-sures 120583

Λand 120583

Λ|xlowastΓΛ

are concentrated on the subset in 119882

lowast

Λ

formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the

sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ

3 The Class of Gibbs States G forthe Fock Space Model

31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G

119911120573for the model

introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882

lowast

Γ the class of these measures will be also denoted byG

Definition 24 Space 119882

lowast

Γis the (infinite) Cartesian product

times119894isinΓ

119882

lowast

119894(cf (60)) its elements are loop configurations Ωlowast

Γ=

Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω

lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2

starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration

Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast

119909119894has finitely many jumps on [0 120573119896

119909119894]

if a jump occurs at time 120591 then

d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)

By W = WΓwe denote the 120590-algebra in 119882

lowast

Γgenerated

by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ

= WΓ

Γthe 120590-subalgebra ofW generated by

cylindrical events localized in Γ Given a probability measure120583 = 120583

Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ

Γthe restriction of

120583 onWΓ

Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub

Γ and Λ

0sube Λ the probability density

119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ

0| Ωlowast0 orΩlowast

ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)

and the conditional partition functions ΞΓ(Λ Λ

0| Ωlowast0 or

ΩlowastΓΛ

) and ΞΓ(Λ | Ωlowast

ΓΛ) are determined as in (96)

As in [1] (110) means that the conditional density119901

Λ0

|ΓΛ(Ωlowast0 | Ωlowast

ΓΛ) relative to 120590-algebra WΓΛ coincides

with 119902

Λ0

ΓΛ(Ωlowast0 | Ωlowast

ΓΛ) for 120583ΓΛ

Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash

aaΩlowast0 isin 119882

lowast

Λ0

Remark 26 The measure 120583Γinherits the property from

Remark 23 and is concentrated on the subset in 119882

lowast

Γformed

by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin

[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ


Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593

120583on the quasilocal 119862lowast-algebra B

First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)

This defines a kernel FΛ0

(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872

lowastΛ0

where Λ

0sub Γ is a finite set of sites It is worth reminding

the reader of the presence of the indicator functionals 1(sdot isinFΛ0

) in (112) and (113) (in the integral for ΞΛ0

Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ

)) These indicators guarantee the compatibilityproperty forall finite Λ0

sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1

(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast

Λ1Λ0)

(114)

Next we identify the operator RΛ0

(a candidate for theRDM in volumeΛ0) as an integral operator acting inH

Λ0 by

(RΛ0

120601) (xlowast0) = int

119872lowastΛ0

FΛ0

(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)

Equation (114) implies that

trHΛ1Λ0

RΛ1

= RΛ0

(116)

Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ

0

If the operators RΛ0

are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)

32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880

(2) fromthe previous section including the hard-core condition for119880

(1)

Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty

Theorem 29 Under condition (22) any FK-DLR state 120593 isin G

is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G

the RDM RΛ0

satisfies (35) Consequently (36) holds true

4 Proof of Theorems 3 6 28 and 29

41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ

0sub Γ

we establish compactness of the sequence of the RDMKsFΛ0

Λ(xlowast0 ylowast0) and FΛ

0

Λ|xlowastΓΛ

(xlowast0 ylowast0) (see (78)ndash(83)) as functions

of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872

lowastΛ0

with

♯xlowast0Λ

= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)

when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ

0

Λand RΛ

0

Λ|xlowastΓΛ

is compact in thetrace-norm operator topology inH

Λ0

To verify compactness of the RDMKs FΛ0


FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing

Lemma 30 (i) Under condition (22) the RDMKs FΛ0

Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0) admit the bounds

FΛ0

Λ(xlowast0 ylowast0) FΛ

0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)

(Note that Φ lt infin under the assumption (22)) Let 119901(119896)

119872yields

the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872

119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)

119872 Finally the upper-bound values 119880(1)

119880

(2) 119869(1) and 119881 have been determined in (14) (15) and (18)


(ii) The gradients of the RDMKs FΛ0


FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)

100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)

(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901

119872

and 119880

(1) 119880(2) 119869(1) and 119881 as in statement (i)

Proof of Lemma 30 First observe that119901119872

lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy

hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0

) + h (ΩlowastΛΛ0 || Ω

lowast0

) (123)

contributing to the RHS in (80) and (81) and the energy

hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)

contributing to the RHS in (82) and (83)This yields the factor

prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)

Next we majorize the integral = int

119882lowast


xlowast0 ylowast0(dΩlowast0

) in(80) and (82) this gives the factor

(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)

The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1

Passing to (121) let us discuss the gradients nabla119909only (The

gradients in the entries of nabla119910are included by symmetry)

The gradient in (121) of course affects only the numeratorsΞΛ0


0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)

and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ

0


the RDMK FΛ0

Λ|xlowastΓΛ

(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast

xlowast0ylowast0(dΩlowast0

) and the other from varying thefunctional exp[minushΛ

0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)]The first contribution can again be uniformly bounded in

terms of the constant 119901119872 The detailed argument as in [1]

includes a deformation of a trajectory and is done similarly

to [1] (the presence of jumps does not change the argumentbecause 119901

119872yields a uniform bound in (43))

The second contribution yields again as in [1] an expres-sion of the form

int

119882lowast



)

times sum

119894isinΛ0

sum

119909isinxlowast0(119894)

h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)

where the functional h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

ΩlowastΛΛ0) is uniformly

bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)

Hence we can guarantee that the RDMs RΛ0

Λand RΛ

0

Λ|xlowastΓΛ

converge to a limiting RDM RΛ0

along a subsequence in Λ

Γ The diagonal process yields convergence for every finiteΛ

0sub Γ A parallel argument leads to compactness of the

measures 120583Λ0

Λfor any given Λ

0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]

In the probabilistic terminology measures 120583Λrepresent

random marked point fields on 119872 times Γ with marks from thespace119882lowast

= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space

of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast

119909119894

introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583

Λ0

Λdescribes the restriction of 120583

Λto volume Λ0 (ie to the

sigma algebra Wlowast

Λ0) and is given by its Radon-Nikodym

derivative 119901

Λ0

Λrelative to the reference measure dΩlowast

Λ0 on

119882

lowast

Λ0 (cf (66) (100)) The reference measure is sigma-finite

Moreover under the condition (22) the value 119901

Λ0

Λ(Ωlowast

Λ0) is

uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882

lowast

Λ0)

This enables us to verify tightness of the family of measures120583

Λ0

Λ Λ Γ and apply the Prokhorov theorem Next we

use the compatibility property of the limit-point measures120583

Λ0

Γand apply the Kolmogorov theorem This establishes the

existence of the limit-point measure 120583Γ

By construction and owing to Lemmas 21 and 22 thelimiting family RΛ

0

yields a state belonging to the class GThis completes the proof of Theorems 3 and 28

42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ

0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin

Λ

0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in


[1] the problem is reduced to checking that forall119911 120573 isin (0infin)

satisfying (22) g isin G and finite Λ0sub Γ

lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| ΩlowastΓΛ(119899)

)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)

here we need to establish this convergence (128) uniformly inthe argumentΩlowast

ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581

and in Ωlowast0 outside a set of the Plowast120573

x0 y0 measure tending to 0

as 119899 rarr infin The latter is formed by path configurations Ωlowast0

that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω

lowast

(119909119894)1205740(119909119894)

119894 isin Λ

0 119909 isin xlowast0 is

defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)

We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy

119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)

As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g

Λ(119899)Λ0 on loop configurations

120596Λ(119899)Λ

0 (over which there is integration performed in thenumerators

ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)

in the expression for 119902

Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0

ΓΛ(119899)(gminus1Ω

lowast0


)) The tuning in gΛ(119899)Λ

0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ

0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall

finite Λ

0sub Γ Ωlowast0 isin 119882

Λ(119899) g isin G and 119886 isin (1infin) for

any 119899 large enough forallΩlowastΛ(119899)Λ

0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and

ΩlowastΓΛ(119899)

= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581

119886

2

exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ

0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2

exp [ minus hΛ(119899)

times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ

0 | Ωlowast

ΓΛ(119899))]

(132)

In (132) the loop configuration gΛ(119899)Λ

0ΩlowastΛ(119899)Λ

0 is deter-mined by specifying its temporal section g

Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ

0 119909 isin xlowast(119894) 0 le 119898 lt 119896

119909119894 That is we

need to specify the sections

(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast

119909119894constituting g

Λ(119899)Λ0Ωlowast

Λ(119899)Λ0 To this end we set

119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)

In other words we apply the action g(119899)119895

to the temporal sec-tions of all loops Ω

lowast

119909119894located at vertex 119895 at a given time

regardless of position of their initial points (119909 119894) inΛ(119899) Λ

0Observe that (130) is deduced from (132) by integrating

in dΩlowastΛ(119899)Λ

0 and normalizing by ΞΛ(119899)Λ

0(ΩlowastΓΛ(119899)

) see (80)with Λ

1015840= Λ(119899) (The Jacobian of the map Ωlowast

Λ(119899)Λ0 997891rarr

gΛ(119899)Λ


0 equals 1)Thus our aim becomes proving (132) The tuned family

gΛ(119899)Λ

0 is composed of individual actions g(119899)119895

isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895

are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579

(119899)

119895denote the vector from119872 corresponding to g(119899)

119895 and we

select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set

120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)

In turn the function 120599 is chosen to satisfy

120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)

Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin

119872

lowastΛ0

the set of path configurations Ωlowast0 isin 119882

lowast

xlowast0 ylowast0 withhΛ0

(Ωlowast0

) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast

x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)


Proof of Lemma 31 The condition that hΛ0

(Ωlowast0

) lt +infin

implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ

0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +

119903(119899))rceil)

Back to the proof of Theorem 29 let gminus1Λ(119899)Λ

0 be the col-lection of the inverse elements

gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)

The vectors corresponding to g(119899)119895

minus1

are minus120579(119899)119895

isin 119872 We will

use this specification for g(119899)119895

and g(119899)119895

minus1

for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)

119895equiv gwhen

119895 isin Λ

0 and g(119899)119895

equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g

Λ(119899)= g(119899)

119895 119895 isin Λ(119899)

Observe that the tuned family gΛ(119899)Λ

0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881

The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)

119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)

Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)

Next the square |120592(119899 119895) minus 120592(119899 119895

1015840)|

2 can be specified as

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))

minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]

2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)

By using convexity of the function exp and (141) forall119886 gt 1

119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2

exp [minushΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

ge 119886 exp [minus12

hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ

0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840

)isinΛ(119899)timesΓ

119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)

The next observation is that

Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))

times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))

minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]

2

(145)

where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)

0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))

minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))

le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)

where function 120577 is determined in (139)


Therefore it remains to estimate the sumsum

119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))

2 To this end observe that 119906120577(119906) lt 1

when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ

119899grows linearly with 119899 Consequently

sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))

997888rarr infin as 119899 997888rarr infin

(148)

Therefore given 119886 gt 1 for 119899 large enough the term119886119890

minus119862Υ2 in the RHS of (145) becomes gt1 Hence

119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)

Equation (149) implies that the quantity

119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0

dΩlowastΛ(119899)Λ

0

times

exp [minushΛ0

(Ωlowast0

orΩlowastΛ(119899)Λ

0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)

(ΩlowastΓΛ(119899)

)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]

ge 2 lim119899rarrinfin

119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)

uniformly in boundary condition 120596ΓΛ(119899)

Integrating (151)d120583ΓΛ(119899)

Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments

This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality

References

[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013

[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010

[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013

[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013

[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998

[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963

[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013

[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782

[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177

[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000

[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977

[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002

[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002

[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011

[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972

[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977

[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979

[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The bi-dimensionality property is expressed in the bound

0 lt sup [1119899

♯Σ (119895 119899) 119895 isin Γ 119899 = 1 2 ] lt infin (2)

where Σ(119895 119899) stands for the set of vertices in Γ at the graphdistance 119899 from 119895 isin Γ (a sphere of radius 119899 around 119895)

Σ (119895 119899) = 119895

1015840isin Γ d (119895 119895

1015840) = 119899 (3)

(The graph distance d(119895 1198951015840) = dΓE(119895 119895

1015840) between 119895 119895

1015840isin Γ is

determined as theminimal length of a path on (ΓE) joining 119895and 1198951015840)This implies that for any 119900 isin Γ the cardinality ♯Λ(119900 119899)of the ball

Λ (119900 119899) = 119895

1015840isin Γ d (119900 119895

1015840) le 119899 (4)

grows at most quadratically with 119899A justification for putting a quantum system on a graph

can be that graph-like structures become increasingly popu-lar in rigorous StatisticalMechanics for example in quantumgravity Namely see [2ndash4] On the other hand a number ofproperties of Gibbs ensembles do not depend upon ldquoregular-ityrdquo of an underlying spatial geometry

12 A BosonicModel in the Fock Space With each vertex 119894 isin Γ

we associate a copy of a compact manifold119872 which we takein this paper to be a unit 119889-dimensional torus R119889

Z119889 witha flat metric 120588 and the volume V We also associate with119894 isin Γ a bosonic Fock-Hilbert space H(119894) ≃ H Here H =

oplus119896=01

H119896whereH

119896= Lsym

2(119872

119896) is the subspace in L

2(119872

119896)

formed by functions symmetric under a permutation of thevariables Given a finite set Λ sub Γ we setH(Λ) = otimes

119894isinΛH(119894)

An element 120601 isin H(Λ) is a complex function

xlowastΛisin 119872

lowastΛ997891997888rarr 120601 (xlowast

Λ) (5)

Here xlowastΛis a collection xlowast(119895) 119895 isin Λ of finite point sets

xlowast(119895) sub 119872 associated with sites 119895 isin Λ Following [1] we callxlowast(119895) a particle configuration at site 119895 (which can be empty)and xlowast

Λa particle configuration in or over Λ The space

119872

lowastΛ of particle configurations inΛ can be represented as theCartesian product (119872lowast

)

timesΛ where 119872

lowast is the disjoint union⋃

119896=01119872

(119896) and 119872

(119896) is the collection of (unordered) 119896-point subsets of119872 (One can consider119872(119896) as the factor of theldquooff-diagonalrdquo set 119872119896

=in the Cartesian power 119872119896 under the

equivalence relation induced by the permutation group oforder 119896) The norm and the scalar product in H

Λare given

by

1003817100381710038171003817

1206011003817100381710038171003817

= (int

119872lowastΛ

1003816100381610038161003816

120601(xlowastΛ)

1003816100381610038161003816

2dxlowastΛ)

12

⟨1206011120601

2⟩ = int

119872lowastΛ

1206011(xlowast

Λ)120601

2(xlowast

Λ)dxlowast

Λ

(6)

where measure dxlowastΛis the product times

119895isinΛdxlowast(119895) and dxlowast(119895) is

the Poissonian sum measure on119872

lowast

dxlowast (119895) = sum

119896=01

1 (♯xlowast (119895) = 119896)

1

119896

prod

119909isinxlowast(119895)dV (119909) 119890minusV(119872)

(7)

Here V(119872) is the volume of torus119872As in [1] we assume that an action

(g 119909) isin G times119872 997891997888rarr g119909 isin 119872 (8)

is given of a group G that is a Euclidean space or a torus ofdimension 119889

1015840le 119889 The action is written as

g119909 = 119909 + 120579119860 mod 1 (9)

Here vector 120579 = (1205791 120579

1198891015840) with components 120579

119897isin [0 1)

and 120579119860 is the 119889-dimensional vector 120579 = ((120579119860)1 (120579119860)

119889)

representing the element g where 119860 is a (1198891015840 times 119889) matrix ofcolumn rank 1198891015840 with rational entries The action of G is liftedto unitary operators U

Λ(g) inH

Λ

UΛ(g)120601 (xlowast

Λ) = 120601 (g

minus1xlowastΛ) (10)

where gminus1xlowastΛ= gminus1xlowast(119895) 119895 isin Λ and gminus1xlowast(119895) = gminus1119909 119909 isin

xlowast(119895)The generally accepted view is that the Hubbard model

is a highly oversimplified model for strongly interactingelectrons in a solidTheHubbardmodel is a kind ofminimummodel which takes into account quantummechanicalmotionof electrons in a solid and nonlinear repulsive interactionbetween electrons There is little doubt that the model is toosimple to describe actual solids faithfully [5] In our contextthe Hubbard Hamiltonian H

Λof the system in Λ acts as

follows

(HΛ120601) (xlowast

Λ) =

[

[

minus

1

2

sum

119895isinΛ

sum


(119909)

119895+ sum

119895isinΛ

sum

119909isinxlowast(119895)119880

(1)(119909)

+

1

2

sum

119895isinΛ

sum

1199091199091015840isinxlowast(119895)

1 (119909 = 119909

1015840)119880

(2)(119909 119909

1015840)

+

1

2

sum

1198951198951015840isinΛ

1 (119895 = 119895

1015840) 119869 (d (119895 119895

1015840))

times sum


119881 (119909 119909

1015840)]

]

120601 (xlowastΛ)

+ sum

1198951198951015840isinΛ


times sum

119909isinxlowast(119895)int

119872

V (d119910)

times [120601 (xlowast(119895119909)rarr (1198951015840

119910)


Λ)]

(11)


HereΔ(119909)




1015840

119910)





ΓΛ= xlowast(1198951015840)

119895

1015840isin Γ Λ isin 119872


ΓΛ


(HΛ|xlowastΓΛ

120601) (xlowastΛ)

=[

[

minus

1

2

sum

119895isinΛ

sum


(119909)

119895+ sum

119895isinΛ

sum

119909isinxlowast(119895)119880

(1)(119909)

+

1

2

sum

119895isinΛ

sum

1199091199091015840isinxlowast(119895)

1 (119909 = 119909

1015840)119880

(2)(119909 119909

1015840)

+

1

2

sum

1198951198951015840isinΛ

1 (119895 = 119895

1015840) 119869 (d (119895 119895

1015840))

times sum


119881 (119909 119909

1015840)]

]

120601 (xlowastΛ)

+ sum

119895isinΛ1198951015840isinΓΛ

119869 (d (119895 1198951015840))

times sum


119881 (119909 119909

1015840)120601 (xlowast

Λ)

+ sum

1198951198951015840isinΛ


times sum


int

119872

V (d119910) [120601 (xlowast(119895119909)rarr (1198951015840

119910)


Λ)]

(12)



1198951198951015840 ge 0 for jumps of a




sup [1205821198951198951015840 (119909119872) 119895 119895



1198951015840d(1198951198951015840)=1 120582119895119895






1015840 119909

10158401015840isin 119872

minus119881(119909

1015840 119909

10158401015840)

10038161003816100381610038161003816

nablax119881(119909

1015840 119909

10158401015840)

10038161003816100381610038161003816

10038161003816100381610038161003816

nablaxx1015840119881(119909

1015840 119909

10158401015840)

10038161003816100381610038161003816

le 119881 (14)


minus119880

(1)(119909)

10038161003816100381610038161003816

nablax119880(1)

(119909)

10038161003816100381610038161003816

le 119880

(1)

119909 isin 119872 (15)



119880

(2)(119909 119909

1015840) = +infin when 1003816

1003816100381610038161003816

119909 minus 119909

101584010038161003816100381610038161003816

le 120588 (16)



1015840) 997891rarr

119880

(2)(119909 119909


119880

(2)(119909 119909

1015840) =

119880

(2)(119909 119909

1015840) whenever 120588(119909 1199091015840) gt 120588 Here 120588(119909 1199091015840)




120581 = lceil

V (119872)

V (119861 (120588))

rceil (17)


minus119880

(2)(119909 119909

1015840)

10038161003816100381610038161003816

nablax119880(2)

(119909 119909

1015840)

10038161003816100381610038161003816

le 119880

(2)

119909 119909

1015840isin 119872 (18)






119869 (119897) = sup[

[

sum

11989510158401015840isinΓ

119869 (d (1198951015840 119895

10158401015840)) 1 (d (1198951015840 11989510158401015840) ge 119897) 119895

1015840isin Γ

]

]

lt infin

(19)


119869

lowast= sup[

[

sum

1198951015840isinΓ

119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

119895 isin Γ]

]

lt infin (20)


119880

(1)(119909) = 119880

(1)(g119909)

119880

(2)(119909 119909

1015840) = 119880

(2)(g119909 g119909

1015840)

119881 (119909 119909

1015840) = 119881 (g119909 g119909

1015840)

(21)


119911119890

Θlt 1 where Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881) (22)


Λ with the action

NΛ120601 (xlowast

Λ) = ♯xlowast

Λ120601 (xlowast

Λ) xlowast

Λisin 119872

lowastΛ (23)




♯xlowastΛ= sum

119895isinΛ

♯xlowast (119895) (24)




Λand H

Λ|xlowastΓΛ


Λ= G

119911120573Λand G

Λ|xlowastΓΛ

= G119911120573Λ|xlowast

ΓΛ

GΛ= 119911


Λ]

GΛ|xlowastΓΛ

= 119911


Λ|xlowastΓΛ

]

(25)




Λ|xlowastΓΛ


Λ|xlowastΓΛ



120573119911Λand 120593

Λ|xΓΛ

= 120593120573119911Λ|x

ΓΛ



spaceHΛ

120593Λ(A) = trH

Λ

(RΛA)

120593Λ|xΓΛ

(A) = trHΛ

(RΛ|xΓΛ

A) A isin BΛ

(26)

where

RΛ=

GΛ


119911120573(Λ) = trH

Λ

GΛ (27)

RΛ|xlowastΓΛ

=

GΛ|xΓΛ


)


) = Ξ119911120573

(Λ | xlowastΓΛ

)

= trHΛ

(119911


Λ|xlowastΓΛ

])

(28)


119911120573(Λ | xlowast




Λ0 is iden-




0

Λof


Λ0 is given by

120593Λ0

Λ(A

0) = trH

Λ0

(RΛ0

ΛA0) A

0isin B

Λ0 (29)

where

RΛ0

Λ= trH

ΛΛ0

RΛ (30)


Operators RΛ0



RΛ0



1sub Λ

RΛ0

Λ= trH

Λ1Λ0

RΛ1

Λ (31)


Λ|xlowastΓΛ

and oper-

ators RΛ0

Λ|xlowastΓΛ





Γ) =


BΛ119899




RΛ1

= trHΛ0Λ1

RΛ0

Λ

1sub Λ

0 (32)



HΛ0

UΛ120601 (xlowast

Λ0) = 120601 (g


where

gminus1xlowast

Λ0 = g


0



(34)




0

Λ





Λ|xΓΛ



for RΛ0

Λ or

RΛ0

Λ|xlowastΓΛ


0)


0 and RΛ0

and RΛ1


0

ΛandRΛ

1

Λor forRΛ

0

Λ|xlowastΓΛ

andRΛ1

Λ|xlowastΓΛ




Λ|xlowastΓΛ




Λ|xlowastΓΛ



for RΛ

0

Λ or RΛ

0

Λ|xlowastΓΛ



UΛ0(g)

minus1RΛ0

UΛ0 (g) = RΛ

0

(35)


0



and g isin G

120593 (A) = 120593 (UΛ0(g)




0


Λ and 120593

Λ How-


Λ|xlowastΓΛ






Λ(xlowast

Λ ylowast

Λ) =

K120573119911Λ

(xlowastΛ ylowast

Λ) and F

Λ(xlowast

Λ ylowast

Λ) = F

120573119911Λ(xlowast

Λ ylowast

Λ) of operators

GΛand R

Λ


(119909119894)(119910119895)denotes



120573



120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)



(37)





=



Λ= ♯ylowast


Λand ylowast

Λis





Λ= ♯ylowast


to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]


119882

120573


119894isinΛ

times

119909isinxlowast(119894)119882

120573

(119909119894)120574(119909119894) (38)


119882

120573

xlowastΛylowastΛ

= ⋃

120574

119882

120573



isin 119882

120573







120573

(119909119894)(119910119895)119882120573


120573

xlowastΛylowastΛ


Λ 997891rarr 120595(119909 119894) by

L120595 (119909 119894) = minus

1

2

Δ120595 (119909 119894)

+ sum

119895d(119894119895)=1

120582119894119895int

119872

V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]

(40)






119894119895V(d119910)




120573

(119909119894)(119910119895)induced




= P120573

(119909119894)(119910119895)(119882

120573

(119909119894)(119910119895)) is given by

119901(119909119894)(119910119895)

= 1 (119894 = 119895) 119901

120573

119872(119909 119910) exp[

[

minus120573 sum

119895d(119894119895)=1

120582119894119895]

]

+ sum

119896ge1

sum

1198970=1198941198971119897119896119897119896+1

=119895

prod

0le119904le119896

1 (d (119897119904 119897119904+1

) = 1)

times 120582119897119904119897119904+1

int

120573

0

d120591119904exp[

[

minus (120591119904+1

minus 120591119904) sum

119895d(119897119904119895)=1

120582119897119904119895]

]

times 1 (0 = 1205910lt 120591


119896lt 120591

119896+1= 120573)

(41)

where 119901120573



119901

120573

119872(119909 119910) =

1

(2120587120573)

1198892

times sum

119899=(1198991119899119889)isinZ119889

exp(minus

1003816100381610038161003816

119909 minus 119910 + 119899

1003816100381610038161003816

2

2120573

)

(42)



119901(119909119894)(119910119895)

10038161003816100381610038161003816

nabla119909119901(119909119894)(119910119895)

10038161003816100381610038161003816

10038161003816100381610038161003816

nabla119910119901(119909119894)(119910119895)

10038161003816100381610038161003816

le 119901119872

119909 119910 isin 119872 119894 119895 isin Γ

(43)

where 119901119872






Λ= ♯ylowast


Λand ylowast

Λ

Then Plowast


120573


P120573


119894isinΛ

times

119909isinxlowast(119894)P120573

(119909119894)120574(119909119894) (44)

Furthermore P120573

xlowastΛylowastΛ


120573

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

= sum

120574

P120573





isin 119882

120573

(119909119894)120574(119909119894)constituting 120596

Λare inde-




120573



Λ) is given by

hΛ (120596Λ) = sum

119894isinΛ

sum

119909isinxlowast(119894)h119909119894 (120596

119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


1015840

1198941015840

)

times (120596119909119894 120596

11990910158401198941015840)

(46)


119909119894 120591) and

11990611990910158401198941015840(120591) = 119906(120596


119909119894isin

119882

120573

(119909119894)120574(119909119894)and 120596

11990910158401198941015840 isin 119882

120573

(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define

h119909119894 (120596119909119894) = int

120573

0

d120591119880(1)(119906

119894119909(120591)) (47)

Next with 119897119909119894(120591) and 119897


119909119894

and 12059611990910158401198941015840 at time 120591

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 120596

11990910158401198941015840)

= int

120573

0

d120591[

[

sum

1198951015840isinΓ

119880

(2)(119906

119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119897

11990910158401198941015840 (120591))

+

1

2

sum

(119895101584011989510158401015840)isinΓtimesΓ

119869 (d (1198951015840 119895

10158401015840))

times 119881 (119906119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119895

10158401015840= 119897

11990910158401198941015840 (120591))

]

]

(48)


ΓΛ) for xlowast

ΓΛ=



ΓΛ) = hΛ (120596

Λ) + hΛ (120596

Λ|| xlowast

ΓΛ) (49)



ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


1015840

1198941015840

)

times (120596119909119894 (119909

1015840 119894

1015840))

(50)

where in turn

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 (119909

1015840 119894

1015840))

= int

120573

0

d120591 [119880(2)(119906

119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119894

1015840)

+ sum

119895isinΓ119895 = 1198941015840

119869 (d (119895 1198941015840))

times119881 (119906119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119895)]

(51)


Λ|| xlowast




Λ(120596

Λ)

120572Λ(120596

Λ) =

1 if index 119897119909119894(120591) isin Λ


(52)



Λand G

Λ|xlowastΓΛ



Λ) = int

119872lowastΛ

prod

119895isinΛ

prod



Λ ylowast

Λ)120601 (ylowast

Λ)

(GΛ|xlowastΓΛ

120601) (xlowastΛ)

= int

119872lowastΛ

prod

119895isinΛ

prod

119910isinylowast(119895)V (d119910)K

Λ(xlowast

Λ ylowast

Λ| xlowast


Λ)

(53)


Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ|

xlowastΓΛ



Λ=


Λ(xlowast

Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ) admit the


KΛ(xlowast

Λ ylowast

Λ) = 119911

♯xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


(54)

KΛ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ)

= 119911

♯ xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


ΓΛ)]

(55)




Λ= 120596

119909119894 such that




Λ= 120596

lowast


of paths 120596119909119894




120573119911(Λ)



Λand trG

Λ|xlowastΓΛ


Λ






lowast


119882

lowast

(119909119894)(119910119895)= ⋃

119896=01

119882

119896120573

(119909119894)(119910119895) (56)



= Ω

lowast

(119909119894)(119910119895)


lowast


lowast


lowast

120591) 119897(Ω

lowast



lowast

(120591 + 120573119898) 0 le 119898 lt 119896(Ω

lowast


(119909119894)(119910119895)isin 119882

lowast

(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))


lowast

119909119894the set 119882lowast

(119909119894)(119909119894) An element of

119882

lowast


lowast



lowast

119909119894isin 119882

lowast


119909119894) or 119896



120591) 119897(Ω



lowast

119909119894(120591 +


(119906(120591 + 120573119898Ω

lowast) 119897(120591 + 120573119898Ω



lowast)]





Λ= ♯ylowast


Λand ylowast

Λ We


119882

lowast


119894isinΛ

times

119909isinxlowast(119894)119882

lowast

(119909119894)120574(119909119894) (57)


119882

lowast

xlowastΛylowastΛ

= ⋃

120574

119882

lowast



lowast




lowast

Λisin 119882

lowast

xlowastΛylowastΛ



xlowastΛ


119882

lowast

xlowastΛ

= times

119894isinΛ

times

119909isinxlowast(119894)119882

lowast

119909119894 (59)

and 119882

lowast


Cartesian power)

119882

lowast

Λ= ⋃


119882

lowast

xlowastΛ

= times

119894isinΛ

119882

lowast

119894

where 119882

lowast

119894= ⋃



lowast

119909119894)

(60)


lowast

Λa collection of



119909119894(120591+120573119898)where 119894 isin Λ



(119909119894)(119910119895) 119882

119909119894 119882lowast


xlowastΛylowastΛ

119882lowast

xlowastΛ

and 119882

lowast

Λ



Λ we comment


lowast

Γof119882lowast




lowast

(119909119894)(119910119895)

Plowast

(119909119894)(119910119895)= sum

119896=01

P119896120573

(119909119894)(119910119895) (61)

Further Plowast


lowast

119909119894

Plowast

119909119894= sum

119896=01

P119896120573

119909119894 (62)





=



Λ= ♯ylowast


Λand ylowast

Λ and we



Plowast


119894isinΛ

times


(119909119894)120574(119909119894) (63)

and the sum measure

Plowast

xlowastΛylowastΛ

= sum

120574

Plowast


Next symbolPlowast

xlowastΛ


xlowastΛ

Plowast

xlowastΛ

= times

119894isinΛ

times


119909119894 (65)


lowast

Λ


Λtimes P

lowast

xlowastΛ

(dΩlowastΛ) (66)

Here for xlowastΛ


=

prod119894isinΛ





Λ| xlowast

ΓΛ) which are


Λ|

xlowastΓΛ


Ω

lowast



hΛ (Ωlowast

Λ)

= sum

119894isinΛ

sum


119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


times 1 ((119909 119894) = (119909

1015840 119894

1015840)) h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

x119894 Ωlowast

11990910158401198941015840)

(67)


119909119894) and h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

119909119894

Ω

lowast

11990910158401198941015840) we set for given 119898 = 0 1 119896

119909119894minus 1 and 119898

1015840=

0 1 11989611990910158401198941015840 minus 1

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

119909119894)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

119909119894)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(68)


index 119897119909119894(120591 + 120573119898Ω

lowast

119909119894) and 119906

119894119909(120591 + 120573119898) for the position

119906(120591 + 120573119898Ω

lowast



lowast

119909119894(120591)

of the loop Ω

lowast


11989410158401199091015840(120591 + 120573119898

1015840)

and 11990611989410158401199091015840(120591 + 120573119898


1015840 119894

1015840)may

coincide) Then

h(119909119894) (Ωlowast

119909119894)

= int

120573

0

d120591[

[

sum

0le119898lt119896119909119894

119880

(1)(119906

119909119894(120591 + 120573119898))

+ sum

0le119898lt1198981015840lt119896119909119894

sum

119895isinΓ

1 (119897119909119894(120591 + 120573119898)

= 119895 = 119897119909119894(120591 + 120573119898

1015840))

times119880

(2)(119906

119909119894(120591 + 120573119898) 119906

119909119894(120591 + 120573119898

1015840))

]

]

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840 )

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(69)



119861 (Ωlowast

Λ) = prod

119894isinΛ

prod


119911

119896119909119894

119896119909119894

(70)



lowast


lowast

119909119894)

arbitrarily



ΓΛ) for xlowast

ΓΛ=


hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)


hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)


Λ|| xlowast

ΓΛ) have a


Λ(Ωlowast



lowast


120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ


0 otherwise(74)



ΓΛ) in (27)


Ξ (Λ) = int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)



Λ= Ω

lowast



119909119894(120591+119898120573))






0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ


Λin (31)

the operator RΛ0


0)

(RΛ0


= int

119872lowastΛ0

prod

119895isinΛ0

prod


0


(77)

where

FΛ0


= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)



ΛΛ0)

=

ΞΛ0


0)

Ξ (Λ)

(78)


Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =




ΛΛ0 denotes


0

Λ|xlowastΓΛ


kernel FΛ0

Λ|xlowastΓΛ


FΛ0

Λ|xlowastΓΛ



0| xlowast

ΓΛ)

Ξ (Λ)

(79)


Λand FΛ

0

Λ|xlowastΓΛ



0) and Ξ

Λ0


0| xlowast

ΓΛ) in





1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










HereΔ(119909)




1015840

119910)





ΓΛ= xlowast(1198951015840)

119895

1015840isin Γ Λ isin 119872


ΓΛ


(HΛ|xlowastΓΛ

120601) (xlowastΛ)

=[

[

minus

1

2

sum

119895isinΛ

sum


(119909)

119895+ sum

119895isinΛ

sum

119909isinxlowast(119895)119880

(1)(119909)

+

1

2

sum

119895isinΛ

sum

1199091199091015840isinxlowast(119895)

1 (119909 = 119909

1015840)119880

(2)(119909 119909

1015840)

+

1

2

sum

1198951198951015840isinΛ

1 (119895 = 119895

1015840) 119869 (d (119895 119895

1015840))

times sum


119881 (119909 119909

1015840)]

]

120601 (xlowastΛ)

+ sum

119895isinΛ1198951015840isinΓΛ

119869 (d (119895 1198951015840))

times sum


119881 (119909 119909

1015840)120601 (xlowast

Λ)

+ sum

1198951198951015840isinΛ


times sum


int

119872

V (d119910) [120601 (xlowast(119895119909)rarr (1198951015840

119910)


Λ)]

(12)



1198951198951015840 ge 0 for jumps of a




sup [1205821198951198951015840 (119909119872) 119895 119895



1198951015840d(1198951198951015840)=1 120582119895119895






1015840 119909

10158401015840isin 119872

minus119881(119909

1015840 119909

10158401015840)

10038161003816100381610038161003816

nablax119881(119909

1015840 119909

10158401015840)

10038161003816100381610038161003816

10038161003816100381610038161003816

nablaxx1015840119881(119909

1015840 119909

10158401015840)

10038161003816100381610038161003816

le 119881 (14)


minus119880

(1)(119909)

10038161003816100381610038161003816

nablax119880(1)

(119909)

10038161003816100381610038161003816

le 119880

(1)

119909 isin 119872 (15)



119880

(2)(119909 119909

1015840) = +infin when 1003816

1003816100381610038161003816

119909 minus 119909

101584010038161003816100381610038161003816

le 120588 (16)



1015840) 997891rarr

119880

(2)(119909 119909


119880

(2)(119909 119909

1015840) =

119880

(2)(119909 119909

1015840) whenever 120588(119909 1199091015840) gt 120588 Here 120588(119909 1199091015840)




120581 = lceil

V (119872)

V (119861 (120588))

rceil (17)


minus119880

(2)(119909 119909

1015840)

10038161003816100381610038161003816

nablax119880(2)

(119909 119909

1015840)

10038161003816100381610038161003816

le 119880

(2)

119909 119909

1015840isin 119872 (18)






119869 (119897) = sup[

[

sum

11989510158401015840isinΓ

119869 (d (1198951015840 119895

10158401015840)) 1 (d (1198951015840 11989510158401015840) ge 119897) 119895

1015840isin Γ

]

]

lt infin

(19)


119869

lowast= sup[

[

sum

1198951015840isinΓ

119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

119895 isin Γ]

]

lt infin (20)


119880

(1)(119909) = 119880

(1)(g119909)

119880

(2)(119909 119909

1015840) = 119880

(2)(g119909 g119909

1015840)

119881 (119909 119909

1015840) = 119881 (g119909 g119909

1015840)

(21)


119911119890

Θlt 1 where Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881) (22)


Λ with the action

NΛ120601 (xlowast

Λ) = ♯xlowast

Λ120601 (xlowast

Λ) xlowast

Λisin 119872

lowastΛ (23)




♯xlowastΛ= sum

119895isinΛ

♯xlowast (119895) (24)




Λand H

Λ|xlowastΓΛ


Λ= G

119911120573Λand G

Λ|xlowastΓΛ

= G119911120573Λ|xlowast

ΓΛ

GΛ= 119911


Λ]

GΛ|xlowastΓΛ

= 119911


Λ|xlowastΓΛ

]

(25)




Λ|xlowastΓΛ


Λ|xlowastΓΛ



120573119911Λand 120593

Λ|xΓΛ

= 120593120573119911Λ|x

ΓΛ



spaceHΛ

120593Λ(A) = trH

Λ

(RΛA)

120593Λ|xΓΛ

(A) = trHΛ

(RΛ|xΓΛ

A) A isin BΛ

(26)

where

RΛ=

GΛ


119911120573(Λ) = trH

Λ

GΛ (27)

RΛ|xlowastΓΛ

=

GΛ|xΓΛ


)


) = Ξ119911120573

(Λ | xlowastΓΛ

)

= trHΛ

(119911


Λ|xlowastΓΛ

])

(28)


119911120573(Λ | xlowast




Λ0 is iden-




0

Λof


Λ0 is given by

120593Λ0

Λ(A

0) = trH

Λ0

(RΛ0

ΛA0) A

0isin B

Λ0 (29)

where

RΛ0

Λ= trH

ΛΛ0

RΛ (30)


Operators RΛ0



RΛ0



1sub Λ

RΛ0

Λ= trH

Λ1Λ0

RΛ1

Λ (31)


Λ|xlowastΓΛ

and oper-

ators RΛ0

Λ|xlowastΓΛ





Γ) =


BΛ119899




RΛ1

= trHΛ0Λ1

RΛ0

Λ

1sub Λ

0 (32)



HΛ0

UΛ120601 (xlowast

Λ0) = 120601 (g


where

gminus1xlowast

Λ0 = g


0



(34)




0

Λ





Λ|xΓΛ



for RΛ0

Λ or

RΛ0

Λ|xlowastΓΛ


0)


0 and RΛ0

and RΛ1


0

ΛandRΛ

1

Λor forRΛ

0

Λ|xlowastΓΛ

andRΛ1

Λ|xlowastΓΛ




Λ|xlowastΓΛ




Λ|xlowastΓΛ



for RΛ

0

Λ or RΛ

0

Λ|xlowastΓΛ



UΛ0(g)

minus1RΛ0

UΛ0 (g) = RΛ

0

(35)


0



and g isin G

120593 (A) = 120593 (UΛ0(g)




0


Λ and 120593

Λ How-


Λ|xlowastΓΛ






Λ(xlowast

Λ ylowast

Λ) =

K120573119911Λ

(xlowastΛ ylowast

Λ) and F

Λ(xlowast

Λ ylowast

Λ) = F

120573119911Λ(xlowast

Λ ylowast

Λ) of operators

GΛand R

Λ


(119909119894)(119910119895)denotes



120573



120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)



(37)





=



Λ= ♯ylowast


Λand ylowast

Λis





Λ= ♯ylowast


to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]


119882

120573


119894isinΛ

times

119909isinxlowast(119894)119882

120573

(119909119894)120574(119909119894) (38)


119882

120573

xlowastΛylowastΛ

= ⋃

120574

119882

120573



isin 119882

120573







120573

(119909119894)(119910119895)119882120573


120573

xlowastΛylowastΛ


Λ 997891rarr 120595(119909 119894) by

L120595 (119909 119894) = minus

1

2

Δ120595 (119909 119894)

+ sum

119895d(119894119895)=1

120582119894119895int

119872

V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]

(40)






119894119895V(d119910)




120573

(119909119894)(119910119895)induced




= P120573

(119909119894)(119910119895)(119882

120573

(119909119894)(119910119895)) is given by

119901(119909119894)(119910119895)

= 1 (119894 = 119895) 119901

120573

119872(119909 119910) exp[

[

minus120573 sum

119895d(119894119895)=1

120582119894119895]

]

+ sum

119896ge1

sum

1198970=1198941198971119897119896119897119896+1

=119895

prod

0le119904le119896

1 (d (119897119904 119897119904+1

) = 1)

times 120582119897119904119897119904+1

int

120573

0

d120591119904exp[

[

minus (120591119904+1

minus 120591119904) sum

119895d(119897119904119895)=1

120582119897119904119895]

]

times 1 (0 = 1205910lt 120591


119896lt 120591

119896+1= 120573)

(41)

where 119901120573



119901

120573

119872(119909 119910) =

1

(2120587120573)

1198892

times sum

119899=(1198991119899119889)isinZ119889

exp(minus

1003816100381610038161003816

119909 minus 119910 + 119899

1003816100381610038161003816

2

2120573

)

(42)



119901(119909119894)(119910119895)

10038161003816100381610038161003816

nabla119909119901(119909119894)(119910119895)

10038161003816100381610038161003816

10038161003816100381610038161003816

nabla119910119901(119909119894)(119910119895)

10038161003816100381610038161003816

le 119901119872

119909 119910 isin 119872 119894 119895 isin Γ

(43)

where 119901119872






Λ= ♯ylowast


Λand ylowast

Λ

Then Plowast


120573


P120573


119894isinΛ

times

119909isinxlowast(119894)P120573

(119909119894)120574(119909119894) (44)

Furthermore P120573

xlowastΛylowastΛ


120573

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

= sum

120574

P120573





isin 119882

120573

(119909119894)120574(119909119894)constituting 120596

Λare inde-




120573



Λ) is given by

hΛ (120596Λ) = sum

119894isinΛ

sum

119909isinxlowast(119894)h119909119894 (120596

119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


1015840

1198941015840

)

times (120596119909119894 120596

11990910158401198941015840)

(46)


119909119894 120591) and

11990611990910158401198941015840(120591) = 119906(120596


119909119894isin

119882

120573

(119909119894)120574(119909119894)and 120596

11990910158401198941015840 isin 119882

120573

(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define

h119909119894 (120596119909119894) = int

120573

0

d120591119880(1)(119906

119894119909(120591)) (47)

Next with 119897119909119894(120591) and 119897


119909119894

and 12059611990910158401198941015840 at time 120591

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 120596

11990910158401198941015840)

= int

120573

0

d120591[

[

sum

1198951015840isinΓ

119880

(2)(119906

119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119897

11990910158401198941015840 (120591))

+

1

2

sum

(119895101584011989510158401015840)isinΓtimesΓ

119869 (d (1198951015840 119895

10158401015840))

times 119881 (119906119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119895

10158401015840= 119897

11990910158401198941015840 (120591))

]

]

(48)


ΓΛ) for xlowast

ΓΛ=



ΓΛ) = hΛ (120596

Λ) + hΛ (120596

Λ|| xlowast

ΓΛ) (49)



ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


1015840

1198941015840

)

times (120596119909119894 (119909

1015840 119894

1015840))

(50)

where in turn

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 (119909

1015840 119894

1015840))

= int

120573

0

d120591 [119880(2)(119906

119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119894

1015840)

+ sum

119895isinΓ119895 = 1198941015840

119869 (d (119895 1198941015840))

times119881 (119906119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119895)]

(51)


Λ|| xlowast




Λ(120596

Λ)

120572Λ(120596

Λ) =

1 if index 119897119909119894(120591) isin Λ


(52)



Λand G

Λ|xlowastΓΛ



Λ) = int

119872lowastΛ

prod

119895isinΛ

prod



Λ ylowast

Λ)120601 (ylowast

Λ)

(GΛ|xlowastΓΛ

120601) (xlowastΛ)

= int

119872lowastΛ

prod

119895isinΛ

prod

119910isinylowast(119895)V (d119910)K

Λ(xlowast

Λ ylowast

Λ| xlowast


Λ)

(53)


Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ|

xlowastΓΛ



Λ=


Λ(xlowast

Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ) admit the


KΛ(xlowast

Λ ylowast

Λ) = 119911

♯xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


(54)

KΛ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ)

= 119911

♯ xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


ΓΛ)]

(55)




Λ= 120596

119909119894 such that




Λ= 120596

lowast


of paths 120596119909119894




120573119911(Λ)



Λand trG

Λ|xlowastΓΛ


Λ






lowast


119882

lowast

(119909119894)(119910119895)= ⋃

119896=01

119882

119896120573

(119909119894)(119910119895) (56)



= Ω

lowast

(119909119894)(119910119895)


lowast


lowast


lowast

120591) 119897(Ω

lowast



lowast

(120591 + 120573119898) 0 le 119898 lt 119896(Ω

lowast


(119909119894)(119910119895)isin 119882

lowast

(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))


lowast

119909119894the set 119882lowast

(119909119894)(119909119894) An element of

119882

lowast


lowast



lowast

119909119894isin 119882

lowast


119909119894) or 119896



120591) 119897(Ω



lowast

119909119894(120591 +


(119906(120591 + 120573119898Ω

lowast) 119897(120591 + 120573119898Ω



lowast)]





Λ= ♯ylowast


Λand ylowast

Λ We


119882

lowast


119894isinΛ

times

119909isinxlowast(119894)119882

lowast

(119909119894)120574(119909119894) (57)


119882

lowast

xlowastΛylowastΛ

= ⋃

120574

119882

lowast



lowast




lowast

Λisin 119882

lowast

xlowastΛylowastΛ



xlowastΛ


119882

lowast

xlowastΛ

= times

119894isinΛ

times

119909isinxlowast(119894)119882

lowast

119909119894 (59)

and 119882

lowast


Cartesian power)

119882

lowast

Λ= ⋃


119882

lowast

xlowastΛ

= times

119894isinΛ

119882

lowast

119894

where 119882

lowast

119894= ⋃



lowast

119909119894)

(60)


lowast

Λa collection of



119909119894(120591+120573119898)where 119894 isin Λ



(119909119894)(119910119895) 119882

119909119894 119882lowast


xlowastΛylowastΛ

119882lowast

xlowastΛ

and 119882

lowast

Λ



Λ we comment


lowast

Γof119882lowast




lowast

(119909119894)(119910119895)

Plowast

(119909119894)(119910119895)= sum

119896=01

P119896120573

(119909119894)(119910119895) (61)

Further Plowast


lowast

119909119894

Plowast

119909119894= sum

119896=01

P119896120573

119909119894 (62)





=



Λ= ♯ylowast


Λand ylowast

Λ and we



Plowast


119894isinΛ

times


(119909119894)120574(119909119894) (63)

and the sum measure

Plowast

xlowastΛylowastΛ

= sum

120574

Plowast


Next symbolPlowast

xlowastΛ


xlowastΛ

Plowast

xlowastΛ

= times

119894isinΛ

times


119909119894 (65)


lowast

Λ


Λtimes P

lowast

xlowastΛ

(dΩlowastΛ) (66)

Here for xlowastΛ


=

prod119894isinΛ





Λ| xlowast

ΓΛ) which are


Λ|

xlowastΓΛ


Ω

lowast



hΛ (Ωlowast

Λ)

= sum

119894isinΛ

sum


119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


times 1 ((119909 119894) = (119909

1015840 119894

1015840)) h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

x119894 Ωlowast

11990910158401198941015840)

(67)


119909119894) and h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

119909119894

Ω

lowast

11990910158401198941015840) we set for given 119898 = 0 1 119896

119909119894minus 1 and 119898

1015840=

0 1 11989611990910158401198941015840 minus 1

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

119909119894)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

119909119894)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(68)


index 119897119909119894(120591 + 120573119898Ω

lowast

119909119894) and 119906

119894119909(120591 + 120573119898) for the position

119906(120591 + 120573119898Ω

lowast



lowast

119909119894(120591)

of the loop Ω

lowast


11989410158401199091015840(120591 + 120573119898

1015840)

and 11990611989410158401199091015840(120591 + 120573119898


1015840 119894

1015840)may

coincide) Then

h(119909119894) (Ωlowast

119909119894)

= int

120573

0

d120591[

[

sum

0le119898lt119896119909119894

119880

(1)(119906

119909119894(120591 + 120573119898))

+ sum

0le119898lt1198981015840lt119896119909119894

sum

119895isinΓ

1 (119897119909119894(120591 + 120573119898)

= 119895 = 119897119909119894(120591 + 120573119898

1015840))

times119880

(2)(119906

119909119894(120591 + 120573119898) 119906

119909119894(120591 + 120573119898

1015840))

]

]

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840 )

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(69)



119861 (Ωlowast

Λ) = prod

119894isinΛ

prod


119911

119896119909119894

119896119909119894

(70)



lowast


lowast

119909119894)

arbitrarily



ΓΛ) for xlowast

ΓΛ=


hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)


hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)


Λ|| xlowast

ΓΛ) have a


Λ(Ωlowast



lowast


120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ


0 otherwise(74)



ΓΛ) in (27)


Ξ (Λ) = int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)



Λ= Ω

lowast



119909119894(120591+119898120573))






0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ


Λin (31)

the operator RΛ0


0)

(RΛ0


= int

119872lowastΛ0

prod

119895isinΛ0

prod


0


(77)

where

FΛ0


= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)



ΛΛ0)

=

ΞΛ0


0)

Ξ (Λ)

(78)


Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =




ΛΛ0 denotes


0

Λ|xlowastΓΛ


kernel FΛ0

Λ|xlowastΓΛ


FΛ0

Λ|xlowastΓΛ



0| xlowast

ΓΛ)

Ξ (Λ)

(79)


Λand FΛ

0

Λ|xlowastΓΛ



0) and Ξ

Λ0


0| xlowast

ΓΛ) in





1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1015840) 997891rarr

119880

(2)(119909 119909


119880

(2)(119909 119909

1015840) =

119880

(2)(119909 119909

1015840) whenever 120588(119909 1199091015840) gt 120588 Here 120588(119909 1199091015840)




120581 = lceil

V (119872)

V (119861 (120588))

rceil (17)


minus119880

(2)(119909 119909

1015840)

10038161003816100381610038161003816

nablax119880(2)

(119909 119909

1015840)

10038161003816100381610038161003816

le 119880

(2)

119909 119909

1015840isin 119872 (18)






119869 (119897) = sup[

[

sum

11989510158401015840isinΓ

119869 (d (1198951015840 119895

10158401015840)) 1 (d (1198951015840 11989510158401015840) ge 119897) 119895

1015840isin Γ

]

]

lt infin

(19)


119869

lowast= sup[

[

sum

1198951015840isinΓ

119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

119895 isin Γ]

]

lt infin (20)


119880

(1)(119909) = 119880

(1)(g119909)

119880

(2)(119909 119909

1015840) = 119880

(2)(g119909 g119909

1015840)

119881 (119909 119909

1015840) = 119881 (g119909 g119909

1015840)

(21)


119911119890

Θlt 1 where Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881) (22)


Λ with the action

NΛ120601 (xlowast

Λ) = ♯xlowast

Λ120601 (xlowast

Λ) xlowast

Λisin 119872

lowastΛ (23)




♯xlowastΛ= sum

119895isinΛ

♯xlowast (119895) (24)




Λand H

Λ|xlowastΓΛ


Λ= G

119911120573Λand G

Λ|xlowastΓΛ

= G119911120573Λ|xlowast

ΓΛ

GΛ= 119911


Λ]

GΛ|xlowastΓΛ

= 119911


Λ|xlowastΓΛ

]

(25)




Λ|xlowastΓΛ


Λ|xlowastΓΛ



120573119911Λand 120593

Λ|xΓΛ

= 120593120573119911Λ|x

ΓΛ



spaceHΛ

120593Λ(A) = trH

Λ

(RΛA)

120593Λ|xΓΛ

(A) = trHΛ

(RΛ|xΓΛ

A) A isin BΛ

(26)

where

RΛ=

GΛ


119911120573(Λ) = trH

Λ

GΛ (27)

RΛ|xlowastΓΛ

=

GΛ|xΓΛ


)


) = Ξ119911120573

(Λ | xlowastΓΛ

)

= trHΛ

(119911


Λ|xlowastΓΛ

])

(28)


119911120573(Λ | xlowast




Λ0 is iden-




0

Λof


Λ0 is given by

120593Λ0

Λ(A

0) = trH

Λ0

(RΛ0

ΛA0) A

0isin B

Λ0 (29)

where

RΛ0

Λ= trH

ΛΛ0

RΛ (30)


Operators RΛ0



RΛ0



1sub Λ

RΛ0

Λ= trH

Λ1Λ0

RΛ1

Λ (31)


Λ|xlowastΓΛ

and oper-

ators RΛ0

Λ|xlowastΓΛ





Γ) =


BΛ119899




RΛ1

= trHΛ0Λ1

RΛ0

Λ

1sub Λ

0 (32)



HΛ0

UΛ120601 (xlowast

Λ0) = 120601 (g


where

gminus1xlowast

Λ0 = g


0



(34)




0

Λ





Λ|xΓΛ



for RΛ0

Λ or

RΛ0

Λ|xlowastΓΛ


0)


0 and RΛ0

and RΛ1


0

ΛandRΛ

1

Λor forRΛ

0

Λ|xlowastΓΛ

andRΛ1

Λ|xlowastΓΛ




Λ|xlowastΓΛ




Λ|xlowastΓΛ



for RΛ

0

Λ or RΛ

0

Λ|xlowastΓΛ



UΛ0(g)

minus1RΛ0

UΛ0 (g) = RΛ

0

(35)


0



and g isin G

120593 (A) = 120593 (UΛ0(g)




0


Λ and 120593

Λ How-


Λ|xlowastΓΛ






Λ(xlowast

Λ ylowast

Λ) =

K120573119911Λ

(xlowastΛ ylowast

Λ) and F

Λ(xlowast

Λ ylowast

Λ) = F

120573119911Λ(xlowast

Λ ylowast

Λ) of operators

GΛand R

Λ


(119909119894)(119910119895)denotes



120573



120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)



(37)





=



Λ= ♯ylowast


Λand ylowast

Λis





Λ= ♯ylowast


to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]


119882

120573


119894isinΛ

times

119909isinxlowast(119894)119882

120573

(119909119894)120574(119909119894) (38)


119882

120573

xlowastΛylowastΛ

= ⋃

120574

119882

120573



isin 119882

120573







120573

(119909119894)(119910119895)119882120573


120573

xlowastΛylowastΛ


Λ 997891rarr 120595(119909 119894) by

L120595 (119909 119894) = minus

1

2

Δ120595 (119909 119894)

+ sum

119895d(119894119895)=1

120582119894119895int

119872

V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]

(40)






119894119895V(d119910)




120573

(119909119894)(119910119895)induced




= P120573

(119909119894)(119910119895)(119882

120573

(119909119894)(119910119895)) is given by

119901(119909119894)(119910119895)

= 1 (119894 = 119895) 119901

120573

119872(119909 119910) exp[

[

minus120573 sum

119895d(119894119895)=1

120582119894119895]

]

+ sum

119896ge1

sum

1198970=1198941198971119897119896119897119896+1

=119895

prod

0le119904le119896

1 (d (119897119904 119897119904+1

) = 1)

times 120582119897119904119897119904+1

int

120573

0

d120591119904exp[

[

minus (120591119904+1

minus 120591119904) sum

119895d(119897119904119895)=1

120582119897119904119895]

]

times 1 (0 = 1205910lt 120591


119896lt 120591

119896+1= 120573)

(41)

where 119901120573



119901

120573

119872(119909 119910) =

1

(2120587120573)

1198892

times sum

119899=(1198991119899119889)isinZ119889

exp(minus

1003816100381610038161003816

119909 minus 119910 + 119899

1003816100381610038161003816

2

2120573

)

(42)



119901(119909119894)(119910119895)

10038161003816100381610038161003816

nabla119909119901(119909119894)(119910119895)

10038161003816100381610038161003816

10038161003816100381610038161003816

nabla119910119901(119909119894)(119910119895)

10038161003816100381610038161003816

le 119901119872

119909 119910 isin 119872 119894 119895 isin Γ

(43)

where 119901119872






Λ= ♯ylowast


Λand ylowast

Λ

Then Plowast


120573


P120573


119894isinΛ

times

119909isinxlowast(119894)P120573

(119909119894)120574(119909119894) (44)

Furthermore P120573

xlowastΛylowastΛ


120573

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

= sum

120574

P120573





isin 119882

120573

(119909119894)120574(119909119894)constituting 120596

Λare inde-




120573



Λ) is given by

hΛ (120596Λ) = sum

119894isinΛ

sum

119909isinxlowast(119894)h119909119894 (120596

119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


1015840

1198941015840

)

times (120596119909119894 120596

11990910158401198941015840)

(46)


119909119894 120591) and

11990611990910158401198941015840(120591) = 119906(120596


119909119894isin

119882

120573

(119909119894)120574(119909119894)and 120596

11990910158401198941015840 isin 119882

120573

(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define

h119909119894 (120596119909119894) = int

120573

0

d120591119880(1)(119906

119894119909(120591)) (47)

Next with 119897119909119894(120591) and 119897


119909119894

and 12059611990910158401198941015840 at time 120591

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 120596

11990910158401198941015840)

= int

120573

0

d120591[

[

sum

1198951015840isinΓ

119880

(2)(119906

119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119897

11990910158401198941015840 (120591))

+

1

2

sum

(119895101584011989510158401015840)isinΓtimesΓ

119869 (d (1198951015840 119895

10158401015840))

times 119881 (119906119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119895

10158401015840= 119897

11990910158401198941015840 (120591))

]

]

(48)


ΓΛ) for xlowast

ΓΛ=



ΓΛ) = hΛ (120596

Λ) + hΛ (120596

Λ|| xlowast

ΓΛ) (49)



ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


1015840

1198941015840

)

times (120596119909119894 (119909

1015840 119894

1015840))

(50)

where in turn

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 (119909

1015840 119894

1015840))

= int

120573

0

d120591 [119880(2)(119906

119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119894

1015840)

+ sum

119895isinΓ119895 = 1198941015840

119869 (d (119895 1198941015840))

times119881 (119906119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119895)]

(51)


Λ|| xlowast




Λ(120596

Λ)

120572Λ(120596

Λ) =

1 if index 119897119909119894(120591) isin Λ


(52)



Λand G

Λ|xlowastΓΛ



Λ) = int

119872lowastΛ

prod

119895isinΛ

prod



Λ ylowast

Λ)120601 (ylowast

Λ)

(GΛ|xlowastΓΛ

120601) (xlowastΛ)

= int

119872lowastΛ

prod

119895isinΛ

prod

119910isinylowast(119895)V (d119910)K

Λ(xlowast

Λ ylowast

Λ| xlowast


Λ)

(53)


Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ|

xlowastΓΛ



Λ=


Λ(xlowast

Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ) admit the


KΛ(xlowast

Λ ylowast

Λ) = 119911

♯xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


(54)

KΛ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ)

= 119911

♯ xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


ΓΛ)]

(55)




Λ= 120596

119909119894 such that




Λ= 120596

lowast


of paths 120596119909119894




120573119911(Λ)



Λand trG

Λ|xlowastΓΛ


Λ






lowast


119882

lowast

(119909119894)(119910119895)= ⋃

119896=01

119882

119896120573

(119909119894)(119910119895) (56)



= Ω

lowast

(119909119894)(119910119895)


lowast


lowast


lowast

120591) 119897(Ω

lowast



lowast

(120591 + 120573119898) 0 le 119898 lt 119896(Ω

lowast


(119909119894)(119910119895)isin 119882

lowast

(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))


lowast

119909119894the set 119882lowast

(119909119894)(119909119894) An element of

119882

lowast


lowast



lowast

119909119894isin 119882

lowast


119909119894) or 119896



120591) 119897(Ω



lowast

119909119894(120591 +


(119906(120591 + 120573119898Ω

lowast) 119897(120591 + 120573119898Ω



lowast)]





Λ= ♯ylowast


Λand ylowast

Λ We


119882

lowast


119894isinΛ

times

119909isinxlowast(119894)119882

lowast

(119909119894)120574(119909119894) (57)


119882

lowast

xlowastΛylowastΛ

= ⋃

120574

119882

lowast



lowast




lowast

Λisin 119882

lowast

xlowastΛylowastΛ



xlowastΛ


119882

lowast

xlowastΛ

= times

119894isinΛ

times

119909isinxlowast(119894)119882

lowast

119909119894 (59)

and 119882

lowast


Cartesian power)

119882

lowast

Λ= ⋃


119882

lowast

xlowastΛ

= times

119894isinΛ

119882

lowast

119894

where 119882

lowast

119894= ⋃



lowast

119909119894)

(60)


lowast

Λa collection of



119909119894(120591+120573119898)where 119894 isin Λ



(119909119894)(119910119895) 119882

119909119894 119882lowast


xlowastΛylowastΛ

119882lowast

xlowastΛ

and 119882

lowast

Λ



Λ we comment


lowast

Γof119882lowast




lowast

(119909119894)(119910119895)

Plowast

(119909119894)(119910119895)= sum

119896=01

P119896120573

(119909119894)(119910119895) (61)

Further Plowast


lowast

119909119894

Plowast

119909119894= sum

119896=01

P119896120573

119909119894 (62)





=



Λ= ♯ylowast


Λand ylowast

Λ and we



Plowast


119894isinΛ

times


(119909119894)120574(119909119894) (63)

and the sum measure

Plowast

xlowastΛylowastΛ

= sum

120574

Plowast


Next symbolPlowast

xlowastΛ


xlowastΛ

Plowast

xlowastΛ

= times

119894isinΛ

times


119909119894 (65)


lowast

Λ


Λtimes P

lowast

xlowastΛ

(dΩlowastΛ) (66)

Here for xlowastΛ


=

prod119894isinΛ





Λ| xlowast

ΓΛ) which are


Λ|

xlowastΓΛ


Ω

lowast



hΛ (Ωlowast

Λ)

= sum

119894isinΛ

sum


119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


times 1 ((119909 119894) = (119909

1015840 119894

1015840)) h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

x119894 Ωlowast

11990910158401198941015840)

(67)


119909119894) and h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

119909119894

Ω

lowast

11990910158401198941015840) we set for given 119898 = 0 1 119896

119909119894minus 1 and 119898

1015840=

0 1 11989611990910158401198941015840 minus 1

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

119909119894)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

119909119894)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(68)


index 119897119909119894(120591 + 120573119898Ω

lowast

119909119894) and 119906

119894119909(120591 + 120573119898) for the position

119906(120591 + 120573119898Ω

lowast



lowast

119909119894(120591)

of the loop Ω

lowast


11989410158401199091015840(120591 + 120573119898

1015840)

and 11990611989410158401199091015840(120591 + 120573119898


1015840 119894

1015840)may

coincide) Then

h(119909119894) (Ωlowast

119909119894)

= int

120573

0

d120591[

[

sum

0le119898lt119896119909119894

119880

(1)(119906

119909119894(120591 + 120573119898))

+ sum

0le119898lt1198981015840lt119896119909119894

sum

119895isinΓ

1 (119897119909119894(120591 + 120573119898)

= 119895 = 119897119909119894(120591 + 120573119898

1015840))

times119880

(2)(119906

119909119894(120591 + 120573119898) 119906

119909119894(120591 + 120573119898

1015840))

]

]

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840 )

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(69)



119861 (Ωlowast

Λ) = prod

119894isinΛ

prod


119911

119896119909119894

119896119909119894

(70)



lowast


lowast

119909119894)

arbitrarily



ΓΛ) for xlowast

ΓΛ=


hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)


hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)


Λ|| xlowast

ΓΛ) have a


Λ(Ωlowast



lowast


120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ


0 otherwise(74)



ΓΛ) in (27)


Ξ (Λ) = int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)



Λ= Ω

lowast



119909119894(120591+119898120573))






0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ


Λin (31)

the operator RΛ0


0)

(RΛ0


= int

119872lowastΛ0

prod

119895isinΛ0

prod


0


(77)

where

FΛ0


= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)



ΛΛ0)

=

ΞΛ0


0)

Ξ (Λ)

(78)


Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =




ΛΛ0 denotes


0

Λ|xlowastΓΛ


kernel FΛ0

Λ|xlowastΓΛ


FΛ0

Λ|xlowastΓΛ



0| xlowast

ΓΛ)

Ξ (Λ)

(79)


Λand FΛ

0

Λ|xlowastΓΛ



0) and Ξ

Λ0


0| xlowast

ΓΛ) in





1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Operators RΛ0



RΛ0



1sub Λ

RΛ0

Λ= trH

Λ1Λ0

RΛ1

Λ (31)


Λ|xlowastΓΛ

and oper-

ators RΛ0

Λ|xlowastΓΛ





Γ) =


BΛ119899




RΛ1

= trHΛ0Λ1

RΛ0

Λ

1sub Λ

0 (32)



HΛ0

UΛ120601 (xlowast

Λ0) = 120601 (g


where

gminus1xlowast

Λ0 = g


0



(34)




0

Λ





Λ|xΓΛ



for RΛ0

Λ or

RΛ0

Λ|xlowastΓΛ


0)


0 and RΛ0

and RΛ1


0

ΛandRΛ

1

Λor forRΛ

0

Λ|xlowastΓΛ

andRΛ1

Λ|xlowastΓΛ




Λ|xlowastΓΛ




Λ|xlowastΓΛ



for RΛ

0

Λ or RΛ

0

Λ|xlowastΓΛ



UΛ0(g)

minus1RΛ0

UΛ0 (g) = RΛ

0

(35)


0



and g isin G

120593 (A) = 120593 (UΛ0(g)




0


Λ and 120593

Λ How-


Λ|xlowastΓΛ






Λ(xlowast

Λ ylowast

Λ) =

K120573119911Λ

(xlowastΛ ylowast

Λ) and F

Λ(xlowast

Λ ylowast

Λ) = F

120573119911Λ(xlowast

Λ ylowast

Λ) of operators

GΛand R

Λ


(119909119894)(119910119895)denotes



120573



120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)



(37)





=



Λ= ♯ylowast


Λand ylowast

Λis





Λ= ♯ylowast


to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]


119882

120573


119894isinΛ

times

119909isinxlowast(119894)119882

120573

(119909119894)120574(119909119894) (38)


119882

120573

xlowastΛylowastΛ

= ⋃

120574

119882

120573



isin 119882

120573







120573

(119909119894)(119910119895)119882120573


120573

xlowastΛylowastΛ


Λ 997891rarr 120595(119909 119894) by

L120595 (119909 119894) = minus

1

2

Δ120595 (119909 119894)

+ sum

119895d(119894119895)=1

120582119894119895int

119872

V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]

(40)






119894119895V(d119910)




120573

(119909119894)(119910119895)induced




= P120573

(119909119894)(119910119895)(119882

120573

(119909119894)(119910119895)) is given by

119901(119909119894)(119910119895)

= 1 (119894 = 119895) 119901

120573

119872(119909 119910) exp[

[

minus120573 sum

119895d(119894119895)=1

120582119894119895]

]

+ sum

119896ge1

sum

1198970=1198941198971119897119896119897119896+1

=119895

prod

0le119904le119896

1 (d (119897119904 119897119904+1

) = 1)

times 120582119897119904119897119904+1

int

120573

0

d120591119904exp[

[

minus (120591119904+1

minus 120591119904) sum

119895d(119897119904119895)=1

120582119897119904119895]

]

times 1 (0 = 1205910lt 120591


119896lt 120591

119896+1= 120573)

(41)

where 119901120573



119901

120573

119872(119909 119910) =

1

(2120587120573)

1198892

times sum

119899=(1198991119899119889)isinZ119889

exp(minus

1003816100381610038161003816

119909 minus 119910 + 119899

1003816100381610038161003816

2

2120573

)

(42)



119901(119909119894)(119910119895)

10038161003816100381610038161003816

nabla119909119901(119909119894)(119910119895)

10038161003816100381610038161003816

10038161003816100381610038161003816

nabla119910119901(119909119894)(119910119895)

10038161003816100381610038161003816

le 119901119872

119909 119910 isin 119872 119894 119895 isin Γ

(43)

where 119901119872






Λ= ♯ylowast


Λand ylowast

Λ

Then Plowast


120573


P120573


119894isinΛ

times

119909isinxlowast(119894)P120573

(119909119894)120574(119909119894) (44)

Furthermore P120573

xlowastΛylowastΛ


120573

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

= sum

120574

P120573





isin 119882

120573

(119909119894)120574(119909119894)constituting 120596

Λare inde-




120573



Λ) is given by

hΛ (120596Λ) = sum

119894isinΛ

sum

119909isinxlowast(119894)h119909119894 (120596

119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


1015840

1198941015840

)

times (120596119909119894 120596

11990910158401198941015840)

(46)


119909119894 120591) and

11990611990910158401198941015840(120591) = 119906(120596


119909119894isin

119882

120573

(119909119894)120574(119909119894)and 120596

11990910158401198941015840 isin 119882

120573

(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define

h119909119894 (120596119909119894) = int

120573

0

d120591119880(1)(119906

119894119909(120591)) (47)

Next with 119897119909119894(120591) and 119897


119909119894

and 12059611990910158401198941015840 at time 120591

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 120596

11990910158401198941015840)

= int

120573

0

d120591[

[

sum

1198951015840isinΓ

119880

(2)(119906

119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119897

11990910158401198941015840 (120591))

+

1

2

sum

(119895101584011989510158401015840)isinΓtimesΓ

119869 (d (1198951015840 119895

10158401015840))

times 119881 (119906119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119895

10158401015840= 119897

11990910158401198941015840 (120591))

]

]

(48)


ΓΛ) for xlowast

ΓΛ=



ΓΛ) = hΛ (120596

Λ) + hΛ (120596

Λ|| xlowast

ΓΛ) (49)



ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


1015840

1198941015840

)

times (120596119909119894 (119909

1015840 119894

1015840))

(50)

where in turn

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 (119909

1015840 119894

1015840))

= int

120573

0

d120591 [119880(2)(119906

119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119894

1015840)

+ sum

119895isinΓ119895 = 1198941015840

119869 (d (119895 1198941015840))

times119881 (119906119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119895)]

(51)


Λ|| xlowast




Λ(120596

Λ)

120572Λ(120596

Λ) =

1 if index 119897119909119894(120591) isin Λ


(52)



Λand G

Λ|xlowastΓΛ



Λ) = int

119872lowastΛ

prod

119895isinΛ

prod



Λ ylowast

Λ)120601 (ylowast

Λ)

(GΛ|xlowastΓΛ

120601) (xlowastΛ)

= int

119872lowastΛ

prod

119895isinΛ

prod

119910isinylowast(119895)V (d119910)K

Λ(xlowast

Λ ylowast

Λ| xlowast


Λ)

(53)


Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ|

xlowastΓΛ



Λ=


Λ(xlowast

Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ) admit the


KΛ(xlowast

Λ ylowast

Λ) = 119911

♯xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


(54)

KΛ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ)

= 119911

♯ xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


ΓΛ)]

(55)




Λ= 120596

119909119894 such that




Λ= 120596

lowast


of paths 120596119909119894




120573119911(Λ)



Λand trG

Λ|xlowastΓΛ


Λ






lowast


119882

lowast

(119909119894)(119910119895)= ⋃

119896=01

119882

119896120573

(119909119894)(119910119895) (56)



= Ω

lowast

(119909119894)(119910119895)


lowast


lowast


lowast

120591) 119897(Ω

lowast



lowast

(120591 + 120573119898) 0 le 119898 lt 119896(Ω

lowast


(119909119894)(119910119895)isin 119882

lowast

(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))


lowast

119909119894the set 119882lowast

(119909119894)(119909119894) An element of

119882

lowast


lowast



lowast

119909119894isin 119882

lowast


119909119894) or 119896



120591) 119897(Ω



lowast

119909119894(120591 +


(119906(120591 + 120573119898Ω

lowast) 119897(120591 + 120573119898Ω



lowast)]





Λ= ♯ylowast


Λand ylowast

Λ We


119882

lowast


119894isinΛ

times

119909isinxlowast(119894)119882

lowast

(119909119894)120574(119909119894) (57)


119882

lowast

xlowastΛylowastΛ

= ⋃

120574

119882

lowast



lowast




lowast

Λisin 119882

lowast

xlowastΛylowastΛ



xlowastΛ


119882

lowast

xlowastΛ

= times

119894isinΛ

times

119909isinxlowast(119894)119882

lowast

119909119894 (59)

and 119882

lowast


Cartesian power)

119882

lowast

Λ= ⋃


119882

lowast

xlowastΛ

= times

119894isinΛ

119882

lowast

119894

where 119882

lowast

119894= ⋃



lowast

119909119894)

(60)


lowast

Λa collection of



119909119894(120591+120573119898)where 119894 isin Λ



(119909119894)(119910119895) 119882

119909119894 119882lowast


xlowastΛylowastΛ

119882lowast

xlowastΛ

and 119882

lowast

Λ



Λ we comment


lowast

Γof119882lowast




lowast

(119909119894)(119910119895)

Plowast

(119909119894)(119910119895)= sum

119896=01

P119896120573

(119909119894)(119910119895) (61)

Further Plowast


lowast

119909119894

Plowast

119909119894= sum

119896=01

P119896120573

119909119894 (62)





=



Λ= ♯ylowast


Λand ylowast

Λ and we



Plowast


119894isinΛ

times


(119909119894)120574(119909119894) (63)

and the sum measure

Plowast

xlowastΛylowastΛ

= sum

120574

Plowast


Next symbolPlowast

xlowastΛ


xlowastΛ

Plowast

xlowastΛ

= times

119894isinΛ

times


119909119894 (65)


lowast

Λ


Λtimes P

lowast

xlowastΛ

(dΩlowastΛ) (66)

Here for xlowastΛ


=

prod119894isinΛ





Λ| xlowast

ΓΛ) which are


Λ|

xlowastΓΛ


Ω

lowast



hΛ (Ωlowast

Λ)

= sum

119894isinΛ

sum


119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


times 1 ((119909 119894) = (119909

1015840 119894

1015840)) h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

x119894 Ωlowast

11990910158401198941015840)

(67)


119909119894) and h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

119909119894

Ω

lowast

11990910158401198941015840) we set for given 119898 = 0 1 119896

119909119894minus 1 and 119898

1015840=

0 1 11989611990910158401198941015840 minus 1

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

119909119894)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

119909119894)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(68)


index 119897119909119894(120591 + 120573119898Ω

lowast

119909119894) and 119906

119894119909(120591 + 120573119898) for the position

119906(120591 + 120573119898Ω

lowast



lowast

119909119894(120591)

of the loop Ω

lowast


11989410158401199091015840(120591 + 120573119898

1015840)

and 11990611989410158401199091015840(120591 + 120573119898


1015840 119894

1015840)may

coincide) Then

h(119909119894) (Ωlowast

119909119894)

= int

120573

0

d120591[

[

sum

0le119898lt119896119909119894

119880

(1)(119906

119909119894(120591 + 120573119898))

+ sum

0le119898lt1198981015840lt119896119909119894

sum

119895isinΓ

1 (119897119909119894(120591 + 120573119898)

= 119895 = 119897119909119894(120591 + 120573119898

1015840))

times119880

(2)(119906

119909119894(120591 + 120573119898) 119906

119909119894(120591 + 120573119898

1015840))

]

]

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840 )

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(69)



119861 (Ωlowast

Λ) = prod

119894isinΛ

prod


119911

119896119909119894

119896119909119894

(70)



lowast


lowast

119909119894)

arbitrarily



ΓΛ) for xlowast

ΓΛ=


hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)


hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)


Λ|| xlowast

ΓΛ) have a


Λ(Ωlowast



lowast


120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ


0 otherwise(74)



ΓΛ) in (27)


Ξ (Λ) = int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)



Λ= Ω

lowast



119909119894(120591+119898120573))






0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ


Λin (31)

the operator RΛ0


0)

(RΛ0


= int

119872lowastΛ0

prod

119895isinΛ0

prod


0


(77)

where

FΛ0


= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)



ΛΛ0)

=

ΞΛ0


0)

Ξ (Λ)

(78)


Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =




ΛΛ0 denotes


0

Λ|xlowastΓΛ


kernel FΛ0

Λ|xlowastΓΛ


FΛ0

Λ|xlowastΓΛ



0| xlowast

ΓΛ)

Ξ (Λ)

(79)


Λand FΛ

0

Λ|xlowastΓΛ



0) and Ξ

Λ0


0| xlowast

ΓΛ) in





1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Λ= ♯ylowast


to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]


119882

120573


119894isinΛ

times

119909isinxlowast(119894)119882

120573

(119909119894)120574(119909119894) (38)


119882

120573

xlowastΛylowastΛ

= ⋃

120574

119882

120573



isin 119882

120573







120573

(119909119894)(119910119895)119882120573


120573

xlowastΛylowastΛ


Λ 997891rarr 120595(119909 119894) by

L120595 (119909 119894) = minus

1

2

Δ120595 (119909 119894)

+ sum

119895d(119894119895)=1

120582119894119895int

119872

V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]

(40)






119894119895V(d119910)




120573

(119909119894)(119910119895)induced




= P120573

(119909119894)(119910119895)(119882

120573

(119909119894)(119910119895)) is given by

119901(119909119894)(119910119895)

= 1 (119894 = 119895) 119901

120573

119872(119909 119910) exp[

[

minus120573 sum

119895d(119894119895)=1

120582119894119895]

]

+ sum

119896ge1

sum

1198970=1198941198971119897119896119897119896+1

=119895

prod

0le119904le119896

1 (d (119897119904 119897119904+1

) = 1)

times 120582119897119904119897119904+1

int

120573

0

d120591119904exp[

[

minus (120591119904+1

minus 120591119904) sum

119895d(119897119904119895)=1

120582119897119904119895]

]

times 1 (0 = 1205910lt 120591


119896lt 120591

119896+1= 120573)

(41)

where 119901120573



119901

120573

119872(119909 119910) =

1

(2120587120573)

1198892

times sum

119899=(1198991119899119889)isinZ119889

exp(minus

1003816100381610038161003816

119909 minus 119910 + 119899

1003816100381610038161003816

2

2120573

)

(42)



119901(119909119894)(119910119895)

10038161003816100381610038161003816

nabla119909119901(119909119894)(119910119895)

10038161003816100381610038161003816

10038161003816100381610038161003816

nabla119910119901(119909119894)(119910119895)

10038161003816100381610038161003816

le 119901119872

119909 119910 isin 119872 119894 119895 isin Γ

(43)

where 119901119872






Λ= ♯ylowast


Λand ylowast

Λ

Then Plowast


120573


P120573


119894isinΛ

times

119909isinxlowast(119894)P120573

(119909119894)120574(119909119894) (44)

Furthermore P120573

xlowastΛylowastΛ


120573

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

= sum

120574

P120573





isin 119882

120573

(119909119894)120574(119909119894)constituting 120596

Λare inde-




120573



Λ) is given by

hΛ (120596Λ) = sum

119894isinΛ

sum

119909isinxlowast(119894)h119909119894 (120596

119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


1015840

1198941015840

)

times (120596119909119894 120596

11990910158401198941015840)

(46)


119909119894 120591) and

11990611990910158401198941015840(120591) = 119906(120596


119909119894isin

119882

120573

(119909119894)120574(119909119894)and 120596

11990910158401198941015840 isin 119882

120573

(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define

h119909119894 (120596119909119894) = int

120573

0

d120591119880(1)(119906

119894119909(120591)) (47)

Next with 119897119909119894(120591) and 119897


119909119894

and 12059611990910158401198941015840 at time 120591

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 120596

11990910158401198941015840)

= int

120573

0

d120591[

[

sum

1198951015840isinΓ

119880

(2)(119906

119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119897

11990910158401198941015840 (120591))

+

1

2

sum

(119895101584011989510158401015840)isinΓtimesΓ

119869 (d (1198951015840 119895

10158401015840))

times 119881 (119906119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119895

10158401015840= 119897

11990910158401198941015840 (120591))

]

]

(48)


ΓΛ) for xlowast

ΓΛ=



ΓΛ) = hΛ (120596

Λ) + hΛ (120596

Λ|| xlowast

ΓΛ) (49)



ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


1015840

1198941015840

)

times (120596119909119894 (119909

1015840 119894

1015840))

(50)

where in turn

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 (119909

1015840 119894

1015840))

= int

120573

0

d120591 [119880(2)(119906

119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119894

1015840)

+ sum

119895isinΓ119895 = 1198941015840

119869 (d (119895 1198941015840))

times119881 (119906119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119895)]

(51)


Λ|| xlowast




Λ(120596

Λ)

120572Λ(120596

Λ) =

1 if index 119897119909119894(120591) isin Λ


(52)



Λand G

Λ|xlowastΓΛ



Λ) = int

119872lowastΛ

prod

119895isinΛ

prod



Λ ylowast

Λ)120601 (ylowast

Λ)

(GΛ|xlowastΓΛ

120601) (xlowastΛ)

= int

119872lowastΛ

prod

119895isinΛ

prod

119910isinylowast(119895)V (d119910)K

Λ(xlowast

Λ ylowast

Λ| xlowast


Λ)

(53)


Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ|

xlowastΓΛ



Λ=


Λ(xlowast

Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ) admit the


KΛ(xlowast

Λ ylowast

Λ) = 119911

♯xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


(54)

KΛ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ)

= 119911

♯ xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


ΓΛ)]

(55)




Λ= 120596

119909119894 such that




Λ= 120596

lowast


of paths 120596119909119894




120573119911(Λ)



Λand trG

Λ|xlowastΓΛ


Λ






lowast


119882

lowast

(119909119894)(119910119895)= ⋃

119896=01

119882

119896120573

(119909119894)(119910119895) (56)



= Ω

lowast

(119909119894)(119910119895)


lowast


lowast


lowast

120591) 119897(Ω

lowast



lowast

(120591 + 120573119898) 0 le 119898 lt 119896(Ω

lowast


(119909119894)(119910119895)isin 119882

lowast

(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))


lowast

119909119894the set 119882lowast

(119909119894)(119909119894) An element of

119882

lowast


lowast



lowast

119909119894isin 119882

lowast


119909119894) or 119896



120591) 119897(Ω



lowast

119909119894(120591 +


(119906(120591 + 120573119898Ω

lowast) 119897(120591 + 120573119898Ω



lowast)]





Λ= ♯ylowast


Λand ylowast

Λ We


119882

lowast


119894isinΛ

times

119909isinxlowast(119894)119882

lowast

(119909119894)120574(119909119894) (57)


119882

lowast

xlowastΛylowastΛ

= ⋃

120574

119882

lowast



lowast




lowast

Λisin 119882

lowast

xlowastΛylowastΛ



xlowastΛ


119882

lowast

xlowastΛ

= times

119894isinΛ

times

119909isinxlowast(119894)119882

lowast

119909119894 (59)

and 119882

lowast


Cartesian power)

119882

lowast

Λ= ⋃


119882

lowast

xlowastΛ

= times

119894isinΛ

119882

lowast

119894

where 119882

lowast

119894= ⋃



lowast

119909119894)

(60)


lowast

Λa collection of



119909119894(120591+120573119898)where 119894 isin Λ



(119909119894)(119910119895) 119882

119909119894 119882lowast


xlowastΛylowastΛ

119882lowast

xlowastΛ

and 119882

lowast

Λ



Λ we comment


lowast

Γof119882lowast




lowast

(119909119894)(119910119895)

Plowast

(119909119894)(119910119895)= sum

119896=01

P119896120573

(119909119894)(119910119895) (61)

Further Plowast


lowast

119909119894

Plowast

119909119894= sum

119896=01

P119896120573

119909119894 (62)





=



Λ= ♯ylowast


Λand ylowast

Λ and we



Plowast


119894isinΛ

times


(119909119894)120574(119909119894) (63)

and the sum measure

Plowast

xlowastΛylowastΛ

= sum

120574

Plowast


Next symbolPlowast

xlowastΛ


xlowastΛ

Plowast

xlowastΛ

= times

119894isinΛ

times


119909119894 (65)


lowast

Λ


Λtimes P

lowast

xlowastΛ

(dΩlowastΛ) (66)

Here for xlowastΛ


=

prod119894isinΛ





Λ| xlowast

ΓΛ) which are


Λ|

xlowastΓΛ


Ω

lowast



hΛ (Ωlowast

Λ)

= sum

119894isinΛ

sum


119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


times 1 ((119909 119894) = (119909

1015840 119894

1015840)) h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

x119894 Ωlowast

11990910158401198941015840)

(67)


119909119894) and h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

119909119894

Ω

lowast

11990910158401198941015840) we set for given 119898 = 0 1 119896

119909119894minus 1 and 119898

1015840=

0 1 11989611990910158401198941015840 minus 1

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

119909119894)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

119909119894)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(68)


index 119897119909119894(120591 + 120573119898Ω

lowast

119909119894) and 119906

119894119909(120591 + 120573119898) for the position

119906(120591 + 120573119898Ω

lowast



lowast

119909119894(120591)

of the loop Ω

lowast


11989410158401199091015840(120591 + 120573119898

1015840)

and 11990611989410158401199091015840(120591 + 120573119898


1015840 119894

1015840)may

coincide) Then

h(119909119894) (Ωlowast

119909119894)

= int

120573

0

d120591[

[

sum

0le119898lt119896119909119894

119880

(1)(119906

119909119894(120591 + 120573119898))

+ sum

0le119898lt1198981015840lt119896119909119894

sum

119895isinΓ

1 (119897119909119894(120591 + 120573119898)

= 119895 = 119897119909119894(120591 + 120573119898

1015840))

times119880

(2)(119906

119909119894(120591 + 120573119898) 119906

119909119894(120591 + 120573119898

1015840))

]

]

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840 )

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(69)



119861 (Ωlowast

Λ) = prod

119894isinΛ

prod


119911

119896119909119894

119896119909119894

(70)



lowast


lowast

119909119894)

arbitrarily



ΓΛ) for xlowast

ΓΛ=


hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)


hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)


Λ|| xlowast

ΓΛ) have a


Λ(Ωlowast



lowast


120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ


0 otherwise(74)



ΓΛ) in (27)


Ξ (Λ) = int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)



Λ= Ω

lowast



119909119894(120591+119898120573))






0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ


Λin (31)

the operator RΛ0


0)

(RΛ0


= int

119872lowastΛ0

prod

119895isinΛ0

prod


0


(77)

where

FΛ0


= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)



ΛΛ0)

=

ΞΛ0


0)

Ξ (Λ)

(78)


Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =




ΛΛ0 denotes


0

Λ|xlowastΓΛ


kernel FΛ0

Λ|xlowastΓΛ


FΛ0

Λ|xlowastΓΛ



0| xlowast

ΓΛ)

Ξ (Λ)

(79)


Λand FΛ

0

Λ|xlowastΓΛ



0) and Ξ

Λ0


0| xlowast

ΓΛ) in





1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120573



Λ) is given by

hΛ (120596Λ) = sum

119894isinΛ

sum

119909isinxlowast(119894)h119909119894 (120596

119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


1015840

1198941015840

)

times (120596119909119894 120596

11990910158401198941015840)

(46)


119909119894 120591) and

11990611990910158401198941015840(120591) = 119906(120596


119909119894isin

119882

120573

(119909119894)120574(119909119894)and 120596

11990910158401198941015840 isin 119882

120573

(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define

h119909119894 (120596119909119894) = int

120573

0

d120591119880(1)(119906

119894119909(120591)) (47)

Next with 119897119909119894(120591) and 119897


119909119894

and 12059611990910158401198941015840 at time 120591

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 120596

11990910158401198941015840)

= int

120573

0

d120591[

[

sum

1198951015840isinΓ

119880

(2)(119906

119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119897

11990910158401198941015840 (120591))

+

1

2

sum

(119895101584011989510158401015840)isinΓtimesΓ

119869 (d (1198951015840 119895

10158401015840))

times 119881 (119906119894119909(120591) 119906

11989410158401199091015840 (120591))

times 1 (119897119909119894(120591) = 119895

1015840= 119895

10158401015840= 119897

11990910158401198941015840 (120591))

]

]

(48)


ΓΛ) for xlowast

ΓΛ=



ΓΛ) = hΛ (120596

Λ) + hΛ (120596

Λ|| xlowast

ΓΛ) (49)



ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


1015840

1198941015840

)

times (120596119909119894 (119909

1015840 119894

1015840))

(50)

where in turn

h(119909119894)(1199091015840

1198941015840

)(120596

119909119894 (119909

1015840 119894

1015840))

= int

120573

0

d120591 [119880(2)(119906

119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119894

1015840)

+ sum

119895isinΓ119895 = 1198941015840

119869 (d (119895 1198941015840))

times119881 (119906119894119909(120591) 119909

1015840) 1 (119897

119909119894(120591) = 119895)]

(51)


Λ|| xlowast




Λ(120596

Λ)

120572Λ(120596

Λ) =

1 if index 119897119909119894(120591) isin Λ


(52)



Λand G

Λ|xlowastΓΛ



Λ) = int

119872lowastΛ

prod

119895isinΛ

prod



Λ ylowast

Λ)120601 (ylowast

Λ)

(GΛ|xlowastΓΛ

120601) (xlowastΛ)

= int

119872lowastΛ

prod

119895isinΛ

prod

119910isinylowast(119895)V (d119910)K

Λ(xlowast

Λ ylowast

Λ| xlowast


Λ)

(53)


Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ|

xlowastΓΛ



Λ=


Λ(xlowast

Λ ylowast

Λ) and K

Λ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ) admit the


KΛ(xlowast

Λ ylowast

Λ) = 119911

♯xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


(54)

KΛ(xlowast

Λ ylowast

Λ| xlowast

ΓΛ)

= 119911

♯ xlowastΛ

int

119882lowast

xlowastΛylowastΛ

P120573

xlowastΛylowastΛ

(d120596Λ) 120572

Λ(120596

Λ)


ΓΛ)]

(55)




Λ= 120596

119909119894 such that




Λ= 120596

lowast


of paths 120596119909119894




120573119911(Λ)



Λand trG

Λ|xlowastΓΛ


Λ






lowast


119882

lowast

(119909119894)(119910119895)= ⋃

119896=01

119882

119896120573

(119909119894)(119910119895) (56)



= Ω

lowast

(119909119894)(119910119895)


lowast


lowast


lowast

120591) 119897(Ω

lowast



lowast

(120591 + 120573119898) 0 le 119898 lt 119896(Ω

lowast


(119909119894)(119910119895)isin 119882

lowast

(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))


lowast

119909119894the set 119882lowast

(119909119894)(119909119894) An element of

119882

lowast


lowast



lowast

119909119894isin 119882

lowast


119909119894) or 119896



120591) 119897(Ω



lowast

119909119894(120591 +


(119906(120591 + 120573119898Ω

lowast) 119897(120591 + 120573119898Ω



lowast)]





Λ= ♯ylowast


Λand ylowast

Λ We


119882

lowast


119894isinΛ

times

119909isinxlowast(119894)119882

lowast

(119909119894)120574(119909119894) (57)


119882

lowast

xlowastΛylowastΛ

= ⋃

120574

119882

lowast



lowast




lowast

Λisin 119882

lowast

xlowastΛylowastΛ



xlowastΛ


119882

lowast

xlowastΛ

= times

119894isinΛ

times

119909isinxlowast(119894)119882

lowast

119909119894 (59)

and 119882

lowast


Cartesian power)

119882

lowast

Λ= ⋃


119882

lowast

xlowastΛ

= times

119894isinΛ

119882

lowast

119894

where 119882

lowast

119894= ⋃



lowast

119909119894)

(60)


lowast

Λa collection of



119909119894(120591+120573119898)where 119894 isin Λ



(119909119894)(119910119895) 119882

119909119894 119882lowast


xlowastΛylowastΛ

119882lowast

xlowastΛ

and 119882

lowast

Λ



Λ we comment


lowast

Γof119882lowast




lowast

(119909119894)(119910119895)

Plowast

(119909119894)(119910119895)= sum

119896=01

P119896120573

(119909119894)(119910119895) (61)

Further Plowast


lowast

119909119894

Plowast

119909119894= sum

119896=01

P119896120573

119909119894 (62)





=



Λ= ♯ylowast


Λand ylowast

Λ and we



Plowast


119894isinΛ

times


(119909119894)120574(119909119894) (63)

and the sum measure

Plowast

xlowastΛylowastΛ

= sum

120574

Plowast


Next symbolPlowast

xlowastΛ


xlowastΛ

Plowast

xlowastΛ

= times

119894isinΛ

times


119909119894 (65)


lowast

Λ


Λtimes P

lowast

xlowastΛ

(dΩlowastΛ) (66)

Here for xlowastΛ


=

prod119894isinΛ





Λ| xlowast

ΓΛ) which are


Λ|

xlowastΓΛ


Ω

lowast



hΛ (Ωlowast

Λ)

= sum

119894isinΛ

sum


119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


times 1 ((119909 119894) = (119909

1015840 119894

1015840)) h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

x119894 Ωlowast

11990910158401198941015840)

(67)


119909119894) and h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

119909119894

Ω

lowast

11990910158401198941015840) we set for given 119898 = 0 1 119896

119909119894minus 1 and 119898

1015840=

0 1 11989611990910158401198941015840 minus 1

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

119909119894)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

119909119894)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(68)


index 119897119909119894(120591 + 120573119898Ω

lowast

119909119894) and 119906

119894119909(120591 + 120573119898) for the position

119906(120591 + 120573119898Ω

lowast



lowast

119909119894(120591)

of the loop Ω

lowast


11989410158401199091015840(120591 + 120573119898

1015840)

and 11990611989410158401199091015840(120591 + 120573119898


1015840 119894

1015840)may

coincide) Then

h(119909119894) (Ωlowast

119909119894)

= int

120573

0

d120591[

[

sum

0le119898lt119896119909119894

119880

(1)(119906

119909119894(120591 + 120573119898))

+ sum

0le119898lt1198981015840lt119896119909119894

sum

119895isinΓ

1 (119897119909119894(120591 + 120573119898)

= 119895 = 119897119909119894(120591 + 120573119898

1015840))

times119880

(2)(119906

119909119894(120591 + 120573119898) 119906

119909119894(120591 + 120573119898

1015840))

]

]

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840 )

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(69)



119861 (Ωlowast

Λ) = prod

119894isinΛ

prod


119911

119896119909119894

119896119909119894

(70)



lowast


lowast

119909119894)

arbitrarily



ΓΛ) for xlowast

ΓΛ=


hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)


hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)


Λ|| xlowast

ΓΛ) have a


Λ(Ωlowast



lowast


120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ


0 otherwise(74)



ΓΛ) in (27)


Ξ (Λ) = int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)



Λ= Ω

lowast



119909119894(120591+119898120573))






0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ


Λin (31)

the operator RΛ0


0)

(RΛ0


= int

119872lowastΛ0

prod

119895isinΛ0

prod


0


(77)

where

FΛ0


= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)



ΛΛ0)

=

ΞΛ0


0)

Ξ (Λ)

(78)


Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =




ΛΛ0 denotes


0

Λ|xlowastΓΛ


kernel FΛ0

Λ|xlowastΓΛ


FΛ0

Λ|xlowastΓΛ



0| xlowast

ΓΛ)

Ξ (Λ)

(79)


Λand FΛ

0

Λ|xlowastΓΛ



0) and Ξ

Λ0


0| xlowast

ΓΛ) in





1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Λ= 120596

119909119894 such that




Λ= 120596

lowast


of paths 120596119909119894




120573119911(Λ)



Λand trG

Λ|xlowastΓΛ


Λ






lowast


119882

lowast

(119909119894)(119910119895)= ⋃

119896=01

119882

119896120573

(119909119894)(119910119895) (56)



= Ω

lowast

(119909119894)(119910119895)


lowast


lowast


lowast

120591) 119897(Ω

lowast



lowast

(120591 + 120573119898) 0 le 119898 lt 119896(Ω

lowast


(119909119894)(119910119895)isin 119882

lowast

(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))


lowast

119909119894the set 119882lowast

(119909119894)(119909119894) An element of

119882

lowast


lowast



lowast

119909119894isin 119882

lowast


119909119894) or 119896



120591) 119897(Ω



lowast

119909119894(120591 +


(119906(120591 + 120573119898Ω

lowast) 119897(120591 + 120573119898Ω



lowast)]





Λ= ♯ylowast


Λand ylowast

Λ We


119882

lowast


119894isinΛ

times

119909isinxlowast(119894)119882

lowast

(119909119894)120574(119909119894) (57)


119882

lowast

xlowastΛylowastΛ

= ⋃

120574

119882

lowast



lowast




lowast

Λisin 119882

lowast

xlowastΛylowastΛ



xlowastΛ


119882

lowast

xlowastΛ

= times

119894isinΛ

times

119909isinxlowast(119894)119882

lowast

119909119894 (59)

and 119882

lowast


Cartesian power)

119882

lowast

Λ= ⋃


119882

lowast

xlowastΛ

= times

119894isinΛ

119882

lowast

119894

where 119882

lowast

119894= ⋃



lowast

119909119894)

(60)


lowast

Λa collection of



119909119894(120591+120573119898)where 119894 isin Λ



(119909119894)(119910119895) 119882

119909119894 119882lowast


xlowastΛylowastΛ

119882lowast

xlowastΛ

and 119882

lowast

Λ



Λ we comment


lowast

Γof119882lowast




lowast

(119909119894)(119910119895)

Plowast

(119909119894)(119910119895)= sum

119896=01

P119896120573

(119909119894)(119910119895) (61)

Further Plowast


lowast

119909119894

Plowast

119909119894= sum

119896=01

P119896120573

119909119894 (62)





=



Λ= ♯ylowast


Λand ylowast

Λ and we



Plowast


119894isinΛ

times


(119909119894)120574(119909119894) (63)

and the sum measure

Plowast

xlowastΛylowastΛ

= sum

120574

Plowast


Next symbolPlowast

xlowastΛ


xlowastΛ

Plowast

xlowastΛ

= times

119894isinΛ

times


119909119894 (65)


lowast

Λ


Λtimes P

lowast

xlowastΛ

(dΩlowastΛ) (66)

Here for xlowastΛ


=

prod119894isinΛ





Λ| xlowast

ΓΛ) which are


Λ|

xlowastΓΛ


Ω

lowast



hΛ (Ωlowast

Λ)

= sum

119894isinΛ

sum


119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


times 1 ((119909 119894) = (119909

1015840 119894

1015840)) h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

x119894 Ωlowast

11990910158401198941015840)

(67)


119909119894) and h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

119909119894

Ω

lowast

11990910158401198941015840) we set for given 119898 = 0 1 119896

119909119894minus 1 and 119898

1015840=

0 1 11989611990910158401198941015840 minus 1

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

119909119894)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

119909119894)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(68)


index 119897119909119894(120591 + 120573119898Ω

lowast

119909119894) and 119906

119894119909(120591 + 120573119898) for the position

119906(120591 + 120573119898Ω

lowast



lowast

119909119894(120591)

of the loop Ω

lowast


11989410158401199091015840(120591 + 120573119898

1015840)

and 11990611989410158401199091015840(120591 + 120573119898


1015840 119894

1015840)may

coincide) Then

h(119909119894) (Ωlowast

119909119894)

= int

120573

0

d120591[

[

sum

0le119898lt119896119909119894

119880

(1)(119906

119909119894(120591 + 120573119898))

+ sum

0le119898lt1198981015840lt119896119909119894

sum

119895isinΓ

1 (119897119909119894(120591 + 120573119898)

= 119895 = 119897119909119894(120591 + 120573119898

1015840))

times119880

(2)(119906

119909119894(120591 + 120573119898) 119906

119909119894(120591 + 120573119898

1015840))

]

]

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840 )

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(69)



119861 (Ωlowast

Λ) = prod

119894isinΛ

prod


119911

119896119909119894

119896119909119894

(70)



lowast


lowast

119909119894)

arbitrarily



ΓΛ) for xlowast

ΓΛ=


hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)


hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)


Λ|| xlowast

ΓΛ) have a


Λ(Ωlowast



lowast


120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ


0 otherwise(74)



ΓΛ) in (27)


Ξ (Λ) = int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)



Λ= Ω

lowast



119909119894(120591+119898120573))






0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ


Λin (31)

the operator RΛ0


0)

(RΛ0


= int

119872lowastΛ0

prod

119895isinΛ0

prod


0


(77)

where

FΛ0


= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)



ΛΛ0)

=

ΞΛ0


0)

Ξ (Λ)

(78)


Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =




ΛΛ0 denotes


0

Λ|xlowastΓΛ


kernel FΛ0

Λ|xlowastΓΛ


FΛ0

Λ|xlowastΓΛ



0| xlowast

ΓΛ)

Ξ (Λ)

(79)


Λand FΛ

0

Λ|xlowastΓΛ



0) and Ξ

Λ0


0| xlowast

ΓΛ) in





1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













=



Λ= ♯ylowast


Λand ylowast

Λ and we



Plowast


119894isinΛ

times


(119909119894)120574(119909119894) (63)

and the sum measure

Plowast

xlowastΛylowastΛ

= sum

120574

Plowast


Next symbolPlowast

xlowastΛ


xlowastΛ

Plowast

xlowastΛ

= times

119894isinΛ

times


119909119894 (65)


lowast

Λ


Λtimes P

lowast

xlowastΛ

(dΩlowastΛ) (66)

Here for xlowastΛ


=

prod119894isinΛ





Λ| xlowast

ΓΛ) which are


Λ|

xlowastΓΛ


Ω

lowast



hΛ (Ωlowast

Λ)

= sum

119894isinΛ

sum


119909119894)

+

1

2

sum

(1198941198941015840)isinΛtimesΛ

sum


times 1 ((119909 119894) = (119909

1015840 119894

1015840)) h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

x119894 Ωlowast

11990910158401198941015840)

(67)


119909119894) and h(119909119894)(119909

1015840

1198941015840

)(Ω

lowast

119909119894

Ω

lowast

11990910158401198941015840) we set for given 119898 = 0 1 119896

119909119894minus 1 and 119898

1015840=

0 1 11989611990910158401198941015840 minus 1

119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

119909119894)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

119909119894)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(68)


index 119897119909119894(120591 + 120573119898Ω

lowast

119909119894) and 119906

119894119909(120591 + 120573119898) for the position

119906(120591 + 120573119898Ω

lowast



lowast

119909119894(120591)

of the loop Ω

lowast


11989410158401199091015840(120591 + 120573119898

1015840)

and 11990611989410158401199091015840(120591 + 120573119898


1015840 119894

1015840)may

coincide) Then

h(119909119894) (Ωlowast

119909119894)

= int

120573

0

d120591[

[

sum

0le119898lt119896119909119894

119880

(1)(119906

119909119894(120591 + 120573119898))

+ sum

0le119898lt1198981015840lt119896119909119894

sum

119895isinΓ

1 (119897119909119894(120591 + 120573119898)

= 119895 = 119897119909119894(120591 + 120573119898

1015840))

times119880

(2)(119906

119909119894(120591 + 120573119898) 119906

119909119894(120591 + 120573119898

1015840))

]

]

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840 )

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(69)



119861 (Ωlowast

Λ) = prod

119894isinΛ

prod


119911

119896119909119894

119896119909119894

(70)



lowast


lowast

119909119894)

arbitrarily



ΓΛ) for xlowast

ΓΛ=


hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)


hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)


Λ|| xlowast

ΓΛ) have a


Λ(Ωlowast



lowast


120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ


0 otherwise(74)



ΓΛ) in (27)


Ξ (Λ) = int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)



Λ= Ω

lowast



119909119894(120591+119898120573))






0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ


Λin (31)

the operator RΛ0


0)

(RΛ0


= int

119872lowastΛ0

prod

119895isinΛ0

prod


0


(77)

where

FΛ0


= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)



ΛΛ0)

=

ΞΛ0


0)

Ξ (Λ)

(78)


Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =




ΛΛ0 denotes


0

Λ|xlowastΓΛ


kernel FΛ0

Λ|xlowastΓΛ


FΛ0

Λ|xlowastΓΛ



0| xlowast

ΓΛ)

Ξ (Λ)

(79)


Λand FΛ

0

Λ|xlowastΓΛ



0) and Ξ

Λ0


0| xlowast

ΓΛ) in





1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ΓΛ) for xlowast

ΓΛ=


hΛ (Ωlowast

Λ| xlowast

ΓΛ) = hΛ (Ω

lowast

Λ) + hΛ (Ω

lowast

Λ|| xlowast

ΓΛ) (71)


hΛ (Ωlowast

Λ|| xlowast

ΓΛ)

= sum

(1198941198941015840)isinΛtimes(ΓΛ)

sum


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(72)

where in turn

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591

times [119880

(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(73)


Λ|| xlowast

ΓΛ) have a


Λ(Ωlowast



lowast


120572Λ(Ω

lowast

Λ) =

1 if Ωlowast

119909119894(120591) isin 119872 times Λ


0 otherwise(74)



ΓΛ) in (27)


Ξ (Λ) = int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ)] (75)


)

= int

119882lowast

Λ


lowast

Λ) 120572

Λ(Ω

lowast


lowast

Λ|| xlowast

ΓΛ)]

(76)



Λ= Ω

lowast



119909119894(120591+119898120573))






0

Λ(see (31)) andRΛ

0

Λ|xlowastΓΛ


Λin (31)

the operator RΛ0


0)

(RΛ0


= int

119872lowastΛ0

prod

119895isinΛ0

prod


0


(77)

where

FΛ0


= int

119872lowastΛΛ0

prod

119895isinΛΛ0

prod

119911isinzlowast(119895)V (d119911)



ΛΛ0)

=

ΞΛ0


0)

Ξ (Λ)

(78)


Λ0 = xlowast(119894) 119894 isin Λ

0 and ylowast

Λ0 =




ΛΛ0 denotes


0

Λ|xlowastΓΛ


kernel FΛ0

Λ|xlowastΓΛ


FΛ0

Λ|xlowastΓΛ



0| xlowast

ΓΛ)

Ξ (Λ)

(79)


Λand FΛ

0

Λ|xlowastΓΛ



0) and Ξ

Λ0


0| xlowast

ΓΛ) in





1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1205740119882

lowast


0


lowast


(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)1205740(119909119894)


lowast

(119909119894)1205740(119909119894)



0(119909 119894)




lowast


xlowast0ylowast0

on119882

lowast

xlowast0 ylowast0


ΛΛ0





0) emerging in


ΞΛ0


0)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

)]ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

(80)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0)

times 120572Λ(Ω

lowast

ΛΛ0) 1 (Ωlowast

ΛΛ0 isin F

Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(81)



0| xlowast

ΓΛ) from (79)


ΞΛ0


0| xlowast

ΓΛ)

= int

119882lowast



) 119861 (Ωlowast0

)

times 120572Λ(Ω

lowast0


)


(Ωlowast0

| xlowastΓΛ

)]

timesΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

(82)

where

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ

)

= int

119882lowast

ΛΛ0


lowast

ΛΛ0) 120572Λ

(Ωlowast

ΛΛ0)


Λ0

)


(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)]

(83)



lowast0

) 1(Ωlowast0 isin

FΛ0


0

)hΛ0

(Ωlowast0

) hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

)hΛ0

(Ωlowast0xlowast

ΓΛ) and hΛΛ

0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ


ΛΛ0) and 120572

Λ(Ωlowast




lowast


lowast


lowast

(119909119894)1205740(119909119894)

isin 119882

lowast

(119909119894)120574lowast0(119909119894)

(Ωlowast


and (119910 119895) = 120574

0(119909 119894) of time-length120573119896

(119909119894)(119910119895)The functional

119861(Ωlowast0

) is given by

119861 (Ωlowast0

) = prod

119894isinΛ

prod

119909isinxlowast(119894)119911

119896(119909119894)(119910119895)

(84)


lowast0


120572Λ(Ω

lowast0

) =

1 if Ωlowast

(119909119894)1205740(119909119894)

(120591) isin 119872 times Λ


(119909119894)(119910119895)

0 otherwise

(85)



0


lowast


(119909119894)(119910119895)


0 and ending up at (119910 119895) = 120574

0(119909 119894) isin

119872timesΛ



for 119897 = 1 119896(119909119894)(119910119895)

minus 1

Ω

lowast

(119909119894)(119910119895)(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

(119909119894)(119910119895)minus 1 (86)

(when 119896(119909119894)(119910119895)



ΛΛ0

is a loop




loops Ωlowast

119909119894 119894 isin Λ Λ


ΛΛ0 isin FΛ

0

) = 1


lowast

119909119894of time-length 120573119896





119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119872times Λ

0 at times 120573119897 for 119897 = 1 119896119909119894

minus 1

Ω

lowast

119909119894(119897120573) notin 119872 times Λ

0 forall119897 = 1 119896

119909119894minus 1 (87)

(again if 119896119909119894


0

(Ωlowast0


0

(ΩlowastΛΛ0 | Ω

lowast0


ΛΛ0 in


hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

) = hΛΛ0

(Ωlowast


ΛΛ0)

(88)





lowast0


lowast

119909119894 isin

119882

lowast


lowast

11990910158401198941015840 isin 119882

lowast

ΛΛ0 we set


= sum

(1198941198941015840

)isinΛ0times(ΛΛ

0

)

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

(89)


119909119894= Ω

lowast

(119909119894)1205740(119909119894)

of time-length 120573119896(119909119894)120574

0(119909119894)

and a loopΩ

lowast

11990910158401198941015840 of time-length 120573119896

119909119894

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896(119909119894)1205740(119909119894)

sum

0le1198981015840lt11989611990910158401198941015840

int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

) 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840))

times 1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

1198951198951015840isinΓtimesΓ

119869 (d (119895 1198951015840))119881 (119906

119894119909(120591 + 120573119898) 119906

11989410158401199091015840 (120591 + 120573119898

1015840))

times1 (119897(119909119894)120574

0(119909119894)

(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]

(90)


(119909119894)1205740(119909119894)

(120591 + 120573119898) andΩ

lowast

11990910158401198941015840(120591 + 120573119898


(119909119894)1205740(119909119894)

and Ω

lowast

11990910158401198941015840 at times 120591 + 120573119898 and

120591 + 120573119898


119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)

11990611989410158401199091015840 (120591 + 120573119898

1015840) = 119906 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

11989711989410158401199091015840 (120591 + 120573119898

1015840) = 119897 (120591 + 120573119898

1015840 Ω

lowast

11990910158401198941015840)

(91)


(Ωlowast0

| xlowastΓΛ


lowast

(119909119894)1205740(119909119894)

instead of Ωlowast

119909119894

Next for hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0

or xlowastΓΛ

) in (83) we set

hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

or xlowastΓΛ

)

= hΛΛ0

(Ωlowast


ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

(92)





lowast0

or xlowastΓΛ


hΛΛ0

(Ωlowast

ΛΛ0 || Ω

lowast0

or xlowastΓΛ

)

= sum

(1198941198941015840)isin(ΛΛ

0

)timesΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

+ sum

119894isinΛΛ0

sum


1015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

(93)

with

h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 Ω

lowast

11990910158401198941015840)

= sum

0le119898lt119896119909119894

sum

0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)

times int

120573

0

d120591

times[

[

sum

119895isinΓ

119880

(2)(119906

119909119894(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

+ sum

(1198951198951015840

)isinΓtimesΓ

119869 (d (119895 1198951015840))119881

times (119906119894119909(120591 + 120573119898) 119906

11990910158401198941015840 (120591 + 120573119898

1015840))

times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895

1015840= 119897

11990910158401198941015840 (120591 + 120573119898

1015840))

]

]


h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










h(119909119894)(1199091015840

1198941015840

)(Ω

lowast

119909119894 (119909

1015840 119894

1015840))

= sum

0le119898lt119896119909119894

int

120573

0

d120591 [119880(2)(119906

119909119894(120591 + 120573119898) 119909

1015840)

times 1 (119897119909119894(120591 + 120573119898) = 119894

1015840)

+ sum

119895isinΓ

119869 (d (119895 1198941015840))119881 (119906

119894119909(120591 + 120573119898) 119909

1015840)

times1 (119897119909119894(120591 + 120573119898) = 119895) ]

(94)


Λ|| xlowast

ΓΛ)



lowast0

ΩlowastΛΛ0)


lowast

(119909119894)1205740(119909119894)

) 119897(120591 + 120573119898Ω

lowast

(119909119894)1205740(119909119894)

)) with 0 le 119898 lt

119896(119909119894)120574

0(119909119894)

119894 isin Λ


1015840 Ω

lowast

11990910158401198941015840) 119897(120591 +

120573119898

1015840 Ω

lowast

11990910158401198941015840)) with 0 le 119898

1015840lt 119896

(11990910158401198941015840) 1198941015840 isin Λ Λ

0 and 119909

1015840isin xlowast(1198941015840)





0

Λ(Λ Λ

0| Ω

lowast0



Λ0

Λ(ΛΛ

0| Ω

lowast0


Λ)


ΛΛ0 isin FΛ

0


Λ


Λ(Λ Λ

0| Ω

lowast0


0

Λ(Λ Λ

0| Ω

lowast0

or xlowastΓΛ




Γ0(Λ |

Ωlowast0


lowast0




2 satisfy

Λ sub Γ

0sube Γ Γ

1 Γ

2sub Γ

Λ

Γ

1cap Γ

2= 0 Λ

0sub Γ (

Λ cup Γ

1cup Γ

2)

(95)

and ♯Λ ♯Λ




over Γ


lowast0

or ΩlowastΓ1) and

ΞΓ0(Λ | Ω

lowast0


Γ2) are given by

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1)]

(96)

ΞΓ0 (

Λ | Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)

= int

119882lowast

Λ


lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| Ω

lowast0

orΩlowast

Γ1 or xlowast

Γ2)]

(97)



119909119894 119894 isin




Γ1 where Ω

lowast0

=

Ω

lowast

(119909119894)1205740(119909119894)

andΩΓ1 = Ω

lowast


1 orΩlowast

Γ1 or xlowast

Γ2






♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

+ ♯ (119909 119894) 119894 isin Λ

0 119897 (120591 + 119898120573Ω

lowast

(119909119894)1205740(119909119894)

)

= 119895 0 le 119898 lt 119896(119909119894)120574

0(119909119894)

+ ♯ (119909 119894 119898) 119894 isin Γ

1

119897 (120591 + 119898120573Ω

lowast

119909119894) = 119895 0 le 119898 lt 119896

119909119894

(98)





Λ|xlowastΓΛ




Λ The sigma


space 119882

lowast



Λ|xlowastΓΛ

are determined



Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Λ|xlowastΓΛ


Λ

119901Λ(Ω

lowast

Λ) =

120583Λ(dΩlowast

Λ)

dΩlowastΛ

=

1

Ξ (Λ)120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ)] Ω

lowast

Λisin 119882

lowast

Λ


(Ωlowast

Λ) =


(dΩlowastΛ)

dΩlowastΛ

1


)

= 120572Λ(Ω

lowast

Λ) 119861 (Ω

lowast

Λ)


Λ| xlowast

ΓΛ)] Ω

lowast

Λisin 119882

lowast

Λ

(99)

Given Λ




Λ

to WΛ0 and 120583

Λ|xlowastΓΛ


Λand 120583Λ

0

Λ|xlowastΓΛ


Λ0

Λ(Ωlowast

Λ0) = 120583Λ

0

Λ(dΩlowast

Λ0)dΩlowast

Λ0 and 119901

Λ0

Λ|xlowastΓΛ

(ΩlowastΛ0) =

120583Λ0

Λ|xlowastΓΛ


Λ0


Λ|xlowastΓΛ



0sub Λ

1015840sub Λ


Λadmits the form

119901

Λ0

Λ(Ω

lowast

Λ0)

= int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840)120583

ΛΛ1015840

Λ(dΩlowast

ΛΛ1015840)

(100)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


0

(Ωlowast

Λ0 | Ω

lowast

ΛΛ1015840)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast

ΛΛ1015840)

ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(101)


1015840 Λ

0| Ωlowast or




Λ|xlowastΓΛ

one has

119901

Λ0

Λ|xlowastΓΛ

(Ωlowast

Λ0) = int

119882lowast

ΛΛ1015840

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

times 120583ΛΛ1015840

Λ|xlowastΓΛ


(102)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast

Λ0 | Ω

lowast


ΓΛ)

= exp [minushΛ0

(Ωlowast

Λ0 | Ω

lowast


ΓΛ)]

times

ΞΛ(Λ

1015840 Λ

0| Ωlowast

Λ0 orΩ

lowast


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(103)


1015840 Λ

0| Ωlowast

Λ0 or


ΓΛ) and Ξ

Λ(Λ

1015840| Ωlowast



(96)


0

Λ(Ωlowast

Λ0 | Ωlowast

ΛΛ1015840) and 119901

Λ0

Λ|xlowastΓΛ





Λ0

ΛΛ1015840(Ω

lowast0|


Λ0

ΛΛ1015840(Ω

lowast0| Ωlowast



1015840

Λmdashand

120583ΛΛ1015840

Λ|xlowastΓΛ


lowast


Λ0 isin 119882

lowast

Λ0


ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast

ΛΛ1015840) and

119902

Λ0

ΛΛ1015840(Ω

lowast

Λ0 | Ωlowast



119902

Λ0

ΛΛ1015840(Ω

lowast0


Λ0

ΛΛ1015840(Ω

lowast0


ΓΛ)


0

ΛΛ1015840(Ω

lowast

Λ0 | Ω

lowast


119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


0


FΛ0

Λ|xlowastΓΛ






forallΛ

0sub Λ

1015840sub Λ

FΛ0


= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

(104)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΛΛ1015840)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ(Λ

1015840| Ωlowast

ΛΛ1015840)

(105)


Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Similarly

FΛ0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 120572Λ(Ω

lowast0

)



)

times int

119882lowast

ΛΛ1015840

120583ΛΛ1015840

Λ|xlowastΓΛ


ΛΛ1015840 isin F

Λ0

)

times 119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast


ΓΛ)

(106)

where

119902

Λ0

ΛΛ1015840 (Ω

lowast0

| Ωlowast

ΛΛ1015840xlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast


ΓΛ)]

times

ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0


ΓΛ)

ΞΛ(Λ

1015840| Ωlowast


ΓΛ)

(107)


Λ(Λ

1015840 Λ

0| Ω

lowast0


ΞΛ0

Λ(Λ

1015840 Λ

0| Ω

lowast0



(83)


Λand 120583

Λ|xlowastΓΛ


lowast

Λ







lowast



lowast


times119894isinΓ

119882

lowast


Γ=


lowast

119909119894 of time-length 120573119896

119909119894where 119896

119909119894= 1 2


Ω

lowast

119909119894 120591 isin [0 120573119896

119909119894] 997891997888rarr (119909 (Ω

lowast

119909119894 120591)

119894 (Ω

lowast

119909119894 120591)) isin 119872 times Γ

Ω

lowast

119909119894is cadlag Ω

lowast

119909119894(0) = Ω

lowast

119909119894(120573119896

119909119894minus) = (119909 119894)

Ω

lowast


119909119894]


d [119894 (Ωlowast

119909119894 120591minus )

119894 (Ω

lowast

119909119894 120591)] = 1

(108)


lowast

Γgenerated


= WΓ



Γon (119882

lowast

ΓW

Γ) we denote by 120583Γ = 120583Γ


120583 onWΓ


Γ and Λ


119901

Λ0

(Ωlowast0) = 119901

Λ0

120583(Ω

lowast0) =

120583Λ0

Γ(dΩlowast0)

^ (dΩlowast0) Ω

lowast0isin 119882

lowast

Λ0

(109)

is of the form

119901

Λ0

(Ωlowast

Λ0) = int

119882lowast

ΓΛ

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)120583

ΓΛ(dΩlowast

ΓΛ) (110)

where

119902

Λ0

ΓΛ(Ω

lowast0| Ω

lowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΓ(Λ Λ


ΓΛ)

ΞΓ(Λ | Ωlowast

ΓΛ)

(111)


0| Ωlowast0 or

ΩlowastΓΛ




Λ0



with 119902

Λ0



Γmdashaa Ωlowast

ΓΛisin 119882

lowast120573

ΓΛand ^

Λ0mdash


lowast

Λ0



lowast

Γformed






First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












First we set

FΛ0

(xlowast0 ylowast0)

= int

119882lowast

x0y0Plowast

x0 y0 (dΩlowast0

) 119861 (Ωlowast0


)

times int

119882lowast

ΓΛ

120583ΓΛ

(dΩlowastΓΛ

) 1 (ΩlowastΓΛ

isin FΛ0

)

times 119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

(112)

where

119902

Λ0

ΓΛ(Ω

lowast0

| Ωlowast

ΓΛ)

= exp [minushΛ0

(Ωlowast0

| Ωlowast

ΓΛ)]

times

ΞΛ0

Λ(Λ Λ

0| Ω

lowast0

orΩlowastΓΛ

)

ΞΓ(Λ | Ωlowast

ΓΛ)

(113)



lowastΛ0

where Λ




Λ(Λ Λ

0|

Ωlowast0

or ΩlowastΓΛ


sub Λ

1

FΛ0

(xlowast0 ylowast0)

= int

119872lowastΛ1Λ0

prod

119895isinΛ1Λ0

prod

119911isinzlowast(119895)V (d119911)

times FΛ1


Λ1Λ0)

(114)



Λ0 by

(RΛ0


119872lowastΛ0

FΛ0



trHΛ1Λ0

RΛ1

= RΛ0

(116)


0





(1)




the RDM RΛ0




0sub Γ



0

Λ|xlowastΓΛ



lowastΛ0

with

♯xlowast0Λ



0

Λand RΛ

0

Λ|xlowastΓΛ


Λ0



FΛ0

Λ|xlowastΓΛ



Λ(xlowast0

ylowast0) and FΛ0

Λ|xlowastΓΛ


FΛ0


0

Λ|xlowastΓΛ

(xlowast0 ylowast0)

le [(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(118)

where

Φ = sum

119896ge1

119911

119896 exp (119896Θ)

with Θ = 120581120573 (119880

(1)

+ 120581119880

(2)

+ 120581119869 (1) 119881)

(119)


119872yields


119901

(119896)

119872= sup

119909119910isin119872

119901

119896120573(119909 119910) = 119901

119896120573(0 0) (120)

and 119901119872

= sup119896ge1

119901

(119896)


119880





FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












FΛ0

Λ|xlowastΓΛ


100381610038161003816100381610038161003816

nabla119909FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119909FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0


100381610038161003816100381610038161003816

100381610038161003816100381610038161003816

nabla119910FΛ0

Λ|xlowastΓΛ

(x0 y0)100381610038161003816100381610038161003816

le (♯Λ

0)ΘΦ

1015840119901119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(121)

where

Φ

1015840= sum

119896ge1

119896119911

119896 exp (119896Θ) (122)


119872

and 119880




hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0) = hΛ

0

(Ωlowast0


lowast0

) (123)


hΛ0

(Ωlowast0

| Ωlowast

ΛΛ0 or xlowast

ΓΛ)

= hΛ0

(Ωlowast0

| xlowastΓΛ


lowast0

)

(124)


prod

120596isinΩlowast0

119911

119896(120596lowast

) exp [119896 (120596lowast)Θ] (125)


119882lowast




(♯Λ

0) 119901

119872[(120581♯Λ

0)] (119901

119872)

120581♯Λ0

Φ

♯Λ0

(126)






0) and Ξ

Λ0


0| xlowast

ΓΛ) in (78)


0


the RDMK FΛ0

Λ|xlowastΓΛ




0

(Ωlowast0

) minus hΛΛ0

(ΩlowastΛΛ0 | Ω

lowast0







int

119882lowast



)

times sum

119894isinΛ0

sum


h119909119894(Ω

lowast

(119909119894)1205740(119909119894)

Ωlowast

ΛΛ0)


(Ωlowast0

) minus hΛΛ0

(Ωlowast

ΛΛ0 | Ω

lowast0

)]

(127)


lowast

(119909119894)1205740(119909119894)




Λand RΛ

0

Λ|xlowastΓΛ





measures 120583Λ0

Λfor any given Λ




= 119882

lowast

0where119882lowast

0= ⋃

119896ge1119882

119896120573

0and119882119896120573

0is the space


119909119894


Λ0





derivative 119901

Λ0


Λ0 on

119882

lowast



Λ0

Λ(Ωlowast

Λ0) is


lowast

Λ0)


Λ0



Λ0




0



0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ

) and 119902

Λ0

ΓΛ(Ω

lowast0

| ΩlowastΓΛ


Λ





lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












lim119899rarrinfin

119902

Λ0

ΓΛ(119899)(gΩ

lowast0


)

119902

Λ0

ΓΛ(119899)(Ω

lowast0


)

= 1 (128)







lowast

(119909119894)1205740(119909119894)

119894 isin Λ


defined by

gΩlowast0

= gΩlowast

(119909119894)1205740(119909119894)

where (gΩlowast

(119909119894)1205740(119909119894)

) (120591) = g (Ωlowast

(119909119894)1205740(119909119894)

(120591))

(129)


119886119902

Λ0

ΓΛ(119899)(gΩ

lowast0

| Ωlowast

ΓΛ(119899)) + 119886119902

Λ0

ΓΛ(119899)(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))

ge 2119902

Λ0

ΓΛ(119899)(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(130)



120596Λ(119899)Λ


ΞΛ0

Γ(Λ (119899) Λ

0| gΩ

lowast0

orΩlowast

ΓΛ(119899))

ΞΛ0

Γ(Λ (119899) Λ

0| g

minus1Ω

lowast0

orΩlowast

ΓΛ(119899))

(131)


Λ0

ΓΛ(119899)(gΩ

lowast0


) and119902

Λ0


lowast0





finite Λ




0 = Ωlowast(119894) 119894 isin Λ(119899) Λ

0 and



119886

2


0Ωlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


times ((gminus1Ω

lowast0

) or (gminus1

Λ(119899)Λ0Ω

lowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(132)




Λ(119899)Λ0Ω

lowast

119909119894(120591 +

120573119898) 119894 isin Λ(119899) Λ


119909119894 That is we


(119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) 119897 (120591 + 120573119898 g

Λ(119899)Λ0Ω

lowast

119909119894)) (133)

for loopsΩlowast




119906 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = g

(119899)

119895119906 (120591 + 120573119898Ω

lowast

119909119894)

if 119897 (120591 + 120573119898 gΛ(119899)Λ

0Ω

lowast

119909119894) = 119895

(134)



lowast







) see (80)with Λ


Λ(119899)Λ0 997891rarr

gΛ(119899)Λ



gΛ(119899)Λ


isin G

gΛ(119899)Λ

0 = g(119899)

119895 119895 isin Λ (119899) Λ

0 (135)

Elements g(119899)119895


(119899)


119895 and we


120579

(119899)

119895= 120579120592 (119899 119895) (136)

where

120592 (119899 119895) =

1 d (119900 119895) le 119903 (119899)

120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)

(137)


120599 (119886 119887) = 1 (119886 le 0) +

1 (0 lt 119886 lt 119887)

119876 (119887)

times int

119887

119886

119911 (119906) d119906 119886 119887 isin R

(138)

with

119876 (119887) = int

119887

0

120577 (119906) d119906 sim log log 119887

where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)

1

119906 ln 119906 119887 gt 0

(139)


119872

lowastΛ0


lowast


(Ωlowast0





(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(Ωlowast0

) lt +infin



119903(119899))rceil)



gminus1

Λ(119899)Λ0 = g

(119899)

119895

minus1

119895 isin Λ (119899) Λ

0 (140)


minus1

are minus120579(119899)119895

isin 119872 We will


and g(119899)119895

minus1


119895equiv gwhen

119895 isin Λ

0 and g(119899)119895


Λ(119899)= g(119899)

119895 119895 isin Λ(119899)




119895 gives

100381610038161003816100381610038161003816

119881 (g(119899)

119895119909 g

(119899)

1198951015840119909

1015840) + 119881(g

(119899)

119895

minus1

119909 g(119899)

1198951015840

minus1

119909

1015840) minus 2119881 (119909 119909

1015840)

100381610038161003816100381610038161003816

le 119862

1003816100381610038161003816

120579

1003816100381610038161003816

210038161003816100381610038161003816

120592 (119899 119895) minus 120592 (119899 119895

1015840)

10038161003816100381610038161003816

2

119881 119909 119909

1015840isin 119872

(141)



1015840)|


10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

=

0

if d (119895 119900) d (1198951015840 119900) le 119903 (119899)

0

if d (119895 119900) d (1198951015840 119900) ge 119899

[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


2

if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899

120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[

120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))

2

if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[

(142)


119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


hΛ(119899)|ΓΛ(119899)

times (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) Ω

lowast

ΓΛ(119899))

minus

1

2

hΛ(119899) (gminus1Λ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))] 119890

minus119862Υ2

(143)

where

Υ = Υ (119899 g) = 120573120581

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840))

10038161003816100381610038161003816

120592(119899 119895) minus 120592(119899 119895

1015840)

10038161003816100381610038161003816

2

(144)


Υ le 3120573120581

21003816100381610038161003816

120579

1003816100381610038161003816

2

times sum

(1198951198951015840


1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895

1015840))



2

(145)


0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))


le d (119895 1198951015840)

120577 (d (119895 119900) minus 119903)

119876 (119899 minus 119903 (119899))

(146)

This yields

Υ le 120573120581

23

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

times sum

(1198951198951015840


119869 (d (119895 1198951015840)) d(119895 119895

1015840)

2

120577(d(119895 0) minus 119903 (119899))

2

le

3

1003816100381610038161003816

120579

1003816100381610038161003816

2

119876(119899 minus 119903(119899))

2

[

[

sup119895isinΓ

sum

1198951015840isinΓ

1198691198951198951015840d(119895 119895

1015840)

2]

]

times sum

119895isinΛ119899+1199030

120577(d (119895 0) minus 119903 (119899))

2

(147)




119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119895isinΛ119899+1199030

120577(d(119895 0) minus 119903(119899))




sum

119895isinΛ119899+1199030

120577(d(119895 119900) minus 119903(119899))

2

= sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

times sum

119895isinΣ119896

120577 (119896 minus 119903 (119899))

le 1198620

sum

1le119896le119899+1199030

120577 (119896 minus 119903 (119899))

le 1198621119876 (119899 + 119903

0minus 119903 (119899))

Υ le

119862

119876 (119899 minus 119903 (119899))


(148)



119886

2

exp [minushΛ(119899) (gΛ(119899)

(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]

+

119886

2


(Ωlowast0

orΩlowast

Λ(119899)Λ0) | Ω

lowast

ΓΛ(119899))]


0 | Ωlowast

ΓΛ(119899))]

(149)


119902

Λ0

|ΓΛ(119899)(Ω

lowast0

| ΩΓΛ(119899)

)

= int

119882Λ(119899)Λ0


0

times

exp [minushΛ0

(Ωlowast0


0 | Ωlowast

ΓΛ0)]

ΞΛ(119899)


)

(150)

obeys

lim119899rarrinfin

[119902

Λ0

|ΓΛ(119899)

120573(gΩ

lowast0

| Ωlowast

ΓΛ(119899))

+119902

Λ0

|ΓΛ(119899)

120573(g

minus1Ω

lowast0

| Ωlowast

ΓΛ(119899))]


119902

Λ0

|ΓΛ(119899)

120573(Ω

lowast0

| Ωlowast

ΓΛ(119899))

(151)



Γ(120596

ΓΛ(119899)) yields (132)

Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Quantum Mermin-Wagner Theorem for a ...

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