A class of infinite dimensional stochastic processes with ...642221/...sional stochastic processes....

Linköping Studies in Science and Technology, Licentiate ThesisNo. 1612

A class of infinite dimensionalstochastic processes with

unbounded diffusion

John Karlsson

Department of MathematicsLinköping University, SE–581 83 Linköping, Sweden

Linköping 2013

Linköping Studies in Science and Technology, Licentiate ThesisNo. 1612

A class of infinite dimensional stochastic processes with unbounded diffusion

John Karlsson

[email protected]

Mathematical StatisticsDepartment of Mathematics

Linköping UniversitySE–581 83 Linköping

Sweden

LIU-TEK-LIC-2013:46ISBN 978-91-7519-536-0

ISSN 0280-7971

Copyright c© 2013 John Karlsson

Printed by LiU-Tryck, Linköping, Sweden 2013, 2nd printing

This is a Swedish Licentiate Thesis. The Licentiate degree comprises 120 ECTS credits ofpostgraduate studies.

Abstract

The aim of this work is to provide an introduction into the theory of infinite dimensionalstochastic processes. The thesis contains the paper A class of infinite dimensional stochas-tic processes with unbounded diffusion written at Linköping University during 2012. Theaim of that paper is to take results from the finite dimensional theory into the infinite di-mensional case. This is done via the means of a coordinate representation. It is shownthat for a certain kind of Dirichlet form with unbounded diffusion, we have propertiessuch as closability, quasi-regularity, and existence of local first and second moment of theassociated process. The starting chapters of this thesis contain the prerequisite theory forunderstanding the paper. It is my hope that any reader unfamiliar with the subject willfind this thesis useful, as an introduction to the field of infinite dimensional processes.

v

Populärvetenskaplig sammanfattning

En stokastisk process är något som beskriver hur ett slumpmässigt system ändrar sig övertiden, t.ex. hur antalet kunder i en affär ändras under dagen eller hur värdet av en aktie för-ändras. Stokastiska processer har studerats i över 100 år och det finns många olika typer avdem. En av de enklaste typerna är Markovprocessen som är en slumpmässig process med”dåligt minne”. Nästa värde som processen har beror bara på dess nuvarande tillstånd.Markovprocesser har studerats sedan början av 1900-talet när A. A. Markov studeradehur konsonanter och vokaler följer efter varandra i texter. En Markovprocess i kontinuer-lig tid (d.v.s. inga hopp) som följer en kontinuerlig kurva kallas för en diffusionsprocess.En process kan ta värden i flera dimensioner, exempelvis kan en partikels bana i rummetsägas vara tredimensionell. Den här avhandlingen behandlar oändligdimensionella pro-cesser d.v.s. processer som tar värden i rum med oändlig dimension. Man har visat att enviss typ av matematiskt objekt, en så kallad Dirichletform, ibland beskriver en sådan pro-cess. Den här avhandlingen studerar en speciell sorts Dirichletform och hittar krav för attvi ska kunna associera den med en process. Det är möjligt att dessa oändligdimensionellaprocesser har viktig tillämpning inom teoretisk fysik.

vii

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Chapter 2-Malliavin Calculus 52.1 General framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Wiener Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Multiple stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Derivative operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Divergence operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7 The generator of the Ornstein-Uhlenbeck semigroup . . . . . . . . . . . 19

3 Chapter 3-Dirichlet form theory 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Semi-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Dirichlet forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Chapter 4-Main results 294.1 Earlier work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Definition of the form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5 Closability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.6 Quasi-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

ix

x Contents

4.7 The generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.8 Local first moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Chapter 5-Notation and basic definitions 475.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Bibliography 51

1Introduction

THE aim of this work is to provide an introduction into the theory of infinite dimen-sional stochastic processes. The thesis contains the paper A class of infinite dimen-sional stochastic processes with unbounded diffusion, a joint work with Jörg-Uwe Löbus,written at Linköping University during 2012. The starting chapters of the thesis con-tain the prerequisite theory for understanding the paper and it is my hope that any readerunfamiliar with the subject will find this useful as an introduction to the field.

1.1 Background

The theory of stochastic processes goes back to the beginning of the 20th century, andin the middle of the 20th century many important concepts were formed. An excellentoverview of this is the article of P.-A. Meyer [6]. Doob worked on Martingales in thetime period 1940-1950 and in 1945 a paper was published detailing the strong Markovproperty. Itô had his first article about the stochastic integral published in 1944 whichclearly has had a large impact on current theory. In 1948 Hille and Yoshida formulatedthe theorem describing the semigroups of continuous operators. This had a large impacton Markov theory since the nature of a Markov process provides a natural connection tosemigroups. Markov processes are of interest due to their natural occurence. As men-tioned, because the evolution of a Markov process only depends on its present state, anyobservation in nature could be considered Markov provided there is sufficient informa-tion. If the time is discrete we call it a Markov chain. While Markov chains changevalue at discrete times a diffusion is a Markov process with continuous trajectories. FromKolmogorov we know that for a Markov process, the transition function is usually a so-lution of the Focker-Planck parabolic partial differential equation. In 1959 Beurling andDeny introduced the theory of Dirichlet forms. Their paper was in the area of potentialtheory. This theory was brought into the realm of probablity by Silverstein in 1974 andFukushima in 1975. This is a useful tool for creating Markov processes in infinite dimen-

1

2 1 Introduction

sional spaces, something that might have applications in physics. In 1976 P. Malliavinintroduced the theory of Malliavin calculus, first developed to prove a theorem in the the-ory of partial differential equations. From the Malliavin calculus comes the Wiener chaosand Ornstein-Uhlenbeck operator which can be used for analysis on the Wiener space andthus allowing analysis on infinite dimensions, once again with possible applications intheoretical physics.

In 1992 B. Driver and M. Röckner [2] published a paper detailing the constructionof an "Ornstein-Uhlenbeck" like process on the path space of a compact Riemannianmanifold without boundary.

In 2004 J.-U. Löbus published a paper [4] describing a class of processes on the pathspace of a Riemannian manifold. In this paper the operator describing the diffusion wasunbounded in contrast to earlier papers.

F.-Y. Wang and B. Wu generalized the results in these papers and showed that theRiemannian manifold can be noncompact, in addition they extended the class of Dirichletforms studied. Their paper [11] was published in 2008. The same authors publishedanother paper [12] in 2009 detailing a large class of quasi-regular Dirichlet forms withunbounded, and non-constant diffusion coefficients on free Riemannian path spaces.

1.2 Outline

This thesis consists of five chapters and the outline is as follows.

Chapter 1

This chapter contains a historical overview of the theory. It also contains the purpose ofthis work as well as providing an outline of the layout of the thesis.

Chapter 2

Chapter two presents the basics of Malliavin Calculus. The chapter begins with introduc-ing the Hermite polynomials as a starting point for the Wiener chaos decomposition. Itis then shown that this decomposition has a representation in the form of Itô integrals.We then present the definition of the Malliavin gradient D and some of its properties,also connecting it to the Wiener chaos decomposition. The end of the chapter contains apresentation of the divergence operator δ, the generator of the Ornstein-Uhlenbeck semi-group L, and the relationship between D, δ and L.

Chapter 3

Chapter three is an introduction to the theory of Dirichlet forms. The first part of thechapter recalls the definitions of resolvents, semigroups, and the infinitesimal generatorassociated to them. We also state the famous Hille–Yosida theorem for contraction semi-groups and by that note the connection between semigroup, resolvent, and generator. Thesecond part of the chapter connects the first part to bilinear forms. The last part of thechapter states the definition of a Dirichlet form and quasi-continuity. We also state the

1.2 Outline 3

theorem connecting the Dirichlet form to a stochastic process. The section ends withstating the famous Cameron–Martin theorem.

Chapter 4

Chapter four contains the paper A class of infinite dimensional stochastic processes withunbounded diffusion written at Linköping University during 2012. The version here ismore detailed than the preprint version at ArXiv. The paper is a study of a particular kindof Dirichlet form. We study the form on cylindrical functions and under some assumptionswe prove the closability of the form. The form is shown to be quasi-regular. The last partsof the paper studies the local first and second moment of the associated process.

Chapter 5

Chapter five contains elementary definitions of concepts used in the thesis. The sectionalso contains a list of notation and abbreviations used throughout this work.

2Chapter 2-Malliavin Calculus

THE Malliavin calculus, named after the French matematician Paul Malliavin, extendsthe calculus of variations, from functions to stochastic processes. In particular itmakes it possible to calculate the derivative of random variables. This section serves tointroduce the notions used in the field of Malliavin calculus. The section is heavily basedon David Nualart’s monographs [8] and [7].

2.1 General framework

Let H be a real separable Hilbert space with inner product 〈·, ·〉H and corresponding norm‖ · ‖H. In this presentation the Hilbert space H will mostly be considered to be L2([0, 1])with the Borel σ-algebra B and Lebesgue measure µ, while in a more general setting itwould be L2(T,B, µ) where (T,B) is a measurable space and µ is an atomless measureon (T,B).

Definition 2.1.1. A stochastic process W = {W (h), h ∈ H} defined in a completeprobability space (Ω,F, P ) is a Gaussian process on H, if W is a centered Gaussianfamily of random variables and E[W (g)W (h)] = 〈g, h〉H for all g, h ∈ H.

This is illustrated by a simple example.

Example 2.1

Let H = L2(R+) and Wt := W (1[0, t]), t ≥ 0. This is standard Brownian motion since

E[WsWt] = E[W (1[0,s])W (1[0,t])] = 〈1[0,s],1[0,t]〉L2 = min(s, t).

Using the same notation as in Nualart [8] letting H1 := W , we note that H1 is a closed

5

6 2 Chapter 2-Malliavin Calculus

subspace of L2(Ω,F, P ), and that W : H → H1 defines a linear isometry between Hand H1. We will denote by W (A) := W (1A). Given a Hilbert space H, a probabilityspace and Gaussian process satisfying the above conditions exist. This is a consequenceof Kolmogorov’s extension theorem.

2.2 Hermite polynomials

The Hermite polynomials will play an important role in the upcoming Wiener chaos de-composition of a Gaussian random variable. The Hermite polynomials Hn are the coeffi-cients of the Taylor expansion of F (x, t) = exp{tx− t2/2} in powers of t, that is:

F (x, t) := etx−t2

2 =

∞∑n=0

Hn(x) · tn. (2.1)

We have the relations:

H0(x) = 1, H1(x) = x, H2(x) =1

2(x2 − 1), Hn(x) =

(−1)n

n!e

x2

2dn

dxn

(e−

x2

2

)and also:

d

dxHn(x) = Hn−1(x), (n+ 1)Hn+1(x) = xHn(x)−Hn−1(x)

as well as:

Hn(0) =

{0 if n odd,(−1)k2kk!

if n even.

Remark 2.2.1. The term Hermite polynomial does not have a strict definition. In otherliterature the polynomials H̃ or Ĥ defined by

etx−t2

2 =

∞∑n=0

H̃n(x) ·tn

n!

or

e2tx−t2

2 =

∞∑n=0

Ĥn(x) ·tn

n!

are classifed as the Hermite polynomials.

2.3 Wiener Chaos

Assume that H (we may think of it asL2([0, 1])) is infinite dimensional and let {e1, e2, . . . }be an ON-basis in H. We let Λ denote the set of all finite multi-indices a = (a1, a2, . . . ),ai ∈ N for i = 1, 2, . . . , i.e. only finitely many ai:s are non-zero. For any a ∈ Λ wedefine:

a! :=

∞∏i=1

ai!, |a| :=∞∑i=1

ai, Φa :=√a!

∞∏i=1

Hai(W (ei)).

Gaussian variables and Hermite polynomials are connected. The next lemma is related toorthogonality in L2(Ω,F, P ).

2.3 Wiener Chaos 7

Lemma 2.3.1. If (X,Y ) is a two-dimensional Gaussian vector s.t. E[X] = E[Y ] = 0and V [X] = V [Y ] = 1. Then for all m,n ∈ N we have:

E[Hm(X)Hn(Y )] =

{0 if m 6= n,1n!E[XY ]

n if m = n.

Proof: For a Gaussian random variable Z we have E[eZ ] = e 12E[Z2] and thus

E[esX+tY ] = e12E[(sX+tY )

2] = e12E[s

2X2]e12E[t

2Y 2]eE[stXY ]

= e12 s

2

e12 t

2

eE[stXY ],

and we get

estE[XY ] = E[esX−s2

2 etY−t2

2 ] = E[(

∞∑m=0

sm(Hm(X))(

∞∑n=0

tnHn(Y ))].

Taylor expansion of the left hand side gives:

estE[XY ] = 1 + stE[XY ] +s2t2E[XY ]2

2!+ · · ·+ s

ntnE[XY ]2

n!+ . . .

and identifying coefficients of smtn yields:

E[Hm(X)Hn(Y )] =

{0 if m 6= n,1n!E[XY ]

n if m = n.

Using this lemma we can proceed to prove that the Hermite polynomials can be usedto construct an ON-basis for L2(Ω,F, P ).

Proposition 2.3.2. The collection {Φa : a ∈ Λ} is an ON-basis for L2(Ω,F, P ).

Proof: Step 1: We begin by showing that {Φa : a ∈ Λ} forms an orthonormal system.Let a, b ∈ Λ. Since W (ei) and W (ej) are independent for i 6= j we get:

E[ΦaΦb] = E[√a!

∞∏i=1

Hai(W (ei)) ·√a!

∞∏i=1

Hbi(W (ei))]

= a! ·∞∏i=1

E[Hai(W (ei))Hbi(W (ei))] =

{1 if a = b,0 otherwise .

Step 2: We show that {Φa : a ∈ Λ} forms a complete system. Assume E[XΦn] = 0 forall a ∈ Λ for some X ∈ L2(Ω,F P ) where X 6= 0. This means that X is orthogonal toevery polynomial in the variables W (e1), . . . ,W (en) for all n ∈ N. We get

E[Xei∑n

j=1 λjW (ej)] = 0, ∀λ1, . . . , λn ∈ R, n ∈ N. (2.2)

The left hand side is the Fourier transform of E[X|W (e1), . . . ,W (en)] and consequentlyE[X|W (e1), . . . ,W (en)] = 0 for all n ∈ N. Since F is generated by W (h), h ∈ Hand e1, e2, . . . is an ON-basis in H it follows that X = 0, thus {Φa : a ∈ Λ} must be acomplete system.


We let Hn denote the closed subspace of L2(Ω,F, P ) spanned by {Φa : a ∈ Λ, |a| =n}. It follows that L2(Ω,F, P ) =

⊕∞n=0 Hn. Hn is called the nth Wiener Chaos. We

denote by Jn the projection operator onto the nth Wiener Chaos, i.e.

F =

∞∑n=0

JnF.

Remark 2.3.3. The nth Wiener chaos contains polynomials of Gaussian variables ofdegree n.

We arrived at the nth Wiener chaos as an expression of our ON-basis (ei)∞i=1. How-ever the following proposition shows that the nth Wiener chaos is independent of thechoice of this ON-basis.

Proposition 2.3.4. For each n ≥ 0 the set of random variables

{Hn(W (h)) : h ∈ H, ‖h‖H = 1}

is dense in Hn.

Proof: The linear span of {Hn(W (h)) : h ∈ H, n ≥ 0} is dense in L2(Ω,F, P ) sinceif a random variable X is orthogonal to this set, it is orthogonal to the span of the sets{W (h)n : h ∈ H, n ≥ 0}. Following the procedure of Proposition 2.3.2 we see that inthis case X = 0. Letting H′n denote the closed span of the family {Hn(W (h)) :, h ∈ H}we get L2(Ω,F, P ) =

⊕∞n=0 H

′n. We have that

⊕Nn=0 H

′n coincides with the set of

polynomials of the family {W (h) : h ∈ H} with degree less than or equal to N . Wealso have that

⊕Nn=0 Hn is the span of the set of polynomials {W (e1),W (e2), . . . } of

degree less than or equal to N . Since (ei)∞i=1 is an ON-basis it follows that⊕∞

n=0 Hn =⊕∞n=0 H

′n and the claim follows.

2.4 Multiple stochastic integrals

We will see that there is another representation for the nth Wiener chaos in the formof iterated Itô integrals. We begin with introducing a type of elementary functions firstdescribed by Norbert Wiener.

Definition 2.4.1. We define En as the set of all elementary functions of form

f(t1, . . . , tn) =

k∑i1,...,in=1

ai1...in1Ai1×···×Ain (t1, . . . , tn), (2.3)

where ai1...in ∈ R and ai1...in = 0 if two indices coincide and A1, . . . , Ak are pairwisedisjoint subsets of [0, 1] with finite Lebesgue measure.

We see that this is the usual simple functions in [0, 1]n with the condition of being0 on any diagonal described by two equal coordinates. It can be shown that this set hasmeasure 0 and we get the following result.

2.4 Multiple stochastic integrals 9

Proposition 2.4.2. The set of elementary functions En is dense in L2([0, 1]n).

Proof: We show that any characteristic function 1A in L2([0, 1]n) whereA = A1×· · ·×An, A1, . . . , An ∈ B([0, 1]) can be approximated by a function in En. Given ε > 0 wecan find B1, . . . , Bk ∈ B([0, 1]) such that |Bi| < ε, i = 1, . . . , k, the sets Bi are disjoint,and every Ai can be written as a union of a collection of the sets Bi. We have

1A =

k∑i1,...,in=1

ai1,...,in1Bi1×···×Bin

where ai1,...,in is either 0 or 1 for each multiindex i. We divide this sum into two parts.We let I be the set of n-tuples (i1, . . . , in) where all the indices are different and J be theremaining set of n-tuples. We define

1B =

k∑(i1,...,in)∈I

ai1,...,in1Bi1×···×Bin .

Now IB belongs to En and we have

‖1A − 1B‖L2([0,1]n) =∥∥∥ ∑

(i1,...,in)∈J

ai1...in1Bi1×···×Bin

∥∥∥L2([0,1]n)

=∑

(i1,...,in)∈J

ai1...inµ(Bi1) · . . . · µ(Bin)

≤(n

2

) k∑i=1

(µ(Bi)

2)( k∑

i=1

µ(Bi))k−2

where the last inequality follows from the facts that(n2

)is loosely the number of diago-

nals, µ(Bi) is the thickness of it and(∑k

i=1 µ(Bi))k−2

is an upper bound of its length.Since Bi is in B[0, 1], i = 1, . . . , k and µ(Bi) < ε we get(

n

2

) k∑i=1

(µ(Bi)

2)( k∑

i=1

µ(Bi))k−2

≤(n

2

)ε( k∑i=1

µ(Bi))k−1

≤(n

2

)ε.

Now since we can approximate characteristic functions it follows that simple functionscan be approximated. As simple functions are dense in L2 the statement follows.

We can now define a stochastic integral for any elementary function. For a functionof the form (2.3) we define

In(f(t1, . . . , tn)) =

k∑i1,...,in=1

ai1,...,inW (Ai1) · . . . ·W (Ain).

Remark 2.4.3. In(f) can be expressed as an iterated Itô integral

In(f) = n!

1∫tn=0

tn∫tn−1=0

. . .

t2∫t1=0

f(t1, . . . , tn) dWt1 . . . dWtn .


The following proposition shows some of the properties of this stochastic integral.

Proposition 2.4.4. In has the following properties:(i) In(f) = In(f̃), where f̃ denotes the symmetrization of f , i.e.

f̃(t1, . . . , tn) =1

n!

∑Π

f(tΠ(1), . . . , tΠ(n))

where Π runs over all permutations of {1, . . . , n}.(ii)

E[Im(f), In(g)] =

{0 if m 6= nn!〈f̃ , g̃〉L2([0,1]n) if m = n.

Proof: (i) Assume that f has the form in (2.3), i.e.

f(t1, . . . , tn) =

k∑i1,...,in=1

ai1...in1Ai1×···×Ain (t1, . . . , tn).

The property follows immediately since

f(tΠ(1), . . . , tΠ(n)) =

k∑i1,...,in=1

aΠ(i1)...Π(in)1AΠ(i1)×···×AΠ(in)(t1, . . . , tn)

=

k∑i1,...,in=1

ai1...in1Ai1×···×Ain (t1, . . . , tn).

(ii) We may assume that f and g are both associated with the same partition A1, . . . , Akotherwise we may refine the partition to satisfy the assumption. Let

g(t1, . . . , tn) :=

k∑i1,...,in=1

bi1...in1Ai1×···×Ain (t1, . . . , tn).

Assume m 6= n. Using (i) we get

E[Im(f)In(g)] = E[Im(f̃)In(g̃)]

= E

[k∑

i1,...,im=1,j1,...,jn=1

ãi1...im b̃j1...jnW (Ai1) · . . . ·W (Ajn)

]= 0

2.5 Derivative operator 11

since at least two of the sets Ai1 . . . Ajn are disjoint. Now for m = n we get

E[In(f)In(g)] = E[Im(f̃)In(g̃)]

= E

[ ∑i1


Remark 2.5.2. We have that DW (h) = h.

Clearly we have the product rule D(FG) = DF · G + F · DG and we also getan integration by parts formula as will be shown shortly. We also define the directionalderivative in the following manner.

Definition 2.5.3. The directional derivative Dh is defined by

DhF := 〈DF, h〉 h ∈ H.

This definition of the gradient D gives the following integration by parts formula

Proposition 2.5.4. Suppose F ∈ Z and h ∈ H. Then E[〈DF, h〉H] = E[F ·W (h)].

Proof: First we assume h = e1 and F = f(W (e1), . . . ,W (en)) where f ∈ C∞p . Nowlet Φ denote the density of a standard n-dimensional normal distribution, i.e.

Φ(x1, . . . , xn) = (2π)−n/2 exp

{−1

2

n∑i=1

x2i

}.

We have

E[〈DF, h〉H] = E[∂f

∂x1(W (e1), . . . ,W (en))] =

∫Rn

∂f

∂x1Φ dx

=

∫Rn

fΦx1 dx = E[F ·W (e1)]) = E[F ·W (h)].

A change of basis proves it in the general case.

We would like to extend D to an operator from L2(Ω,F, P ) to L2(Ω,F, P ;H) anddo so by first stating the next lemma.

Lemma 2.5.5. Let F,G ∈ Z and h ∈ H. Then

E[G〈DF, h〉H] = E[−F 〈DG,h〉H + FGW (h)].

Proof: Using the product rule and Proposition 2.5.4 we get:

E[G〈DF, h〉H] = E[〈G ·DF, h〉H] = E[〈D(FG)− F ·DG,h〉H]= E[−F 〈DG,h〉H] + E[〈D(FG), h〉H]= E[−F 〈DG,h〉H] + E[FG ·W (h)].

Using this lemma we prove the following proposition.

Proposition 2.5.6. The operator D is closable from Lp(Ω,F, P ) to Lp(Ω,F, P ;H) forany p ≥ 1.

Proof: Let Fn ∈ Z, n = 1, 2, . . . such that:

2.5 Derivative operator 13

• Fn → 0 in Lp,

• DFn is Cauchy in Lp(Ω,F, P ;H).

Let η := limn→∞DFn. Left to prove is η = 0. Let h ∈ H andG = g(W (h1), . . . ,W (hn)),g ∈ C∞b (Rn) be any smooth random variable s.t. GW (h) is bounded. These G are densein Lp(Ω,F, P ;H). It follows using the above lemma

E[〈η, h〉HG] = limn→∞

E[〈DFn, h〉HG]

= limn→∞

E[−Fn〈DG,h〉H + FnGW (h)] = 0.

It follows 〈η, h〉H = 0 and since h was arbitrary we get η = 0.

As a consequence there exists a unique closed extension (D,D1,p) of (D,Z). Herethe case p = 2 is mostly considered. The closability of D gives rise to the chain rule

Proposition 2.5.7. (Chain rule) Let ϕ : Rm → R be a continuously differentiable func-tion with bounded partial derivatives. Suppose F = (F1, . . . , Fm) ∈ (D1,p)m for somep ≥ 1. Then ϕ(F ) ∈ D1,p and

D(ϕ(F )) =

m∑i=1

∂ϕ

∂xi(F )DFi.

Proof: Since D1,p is closed we can find sequences F (n)i ∈ Z, i = 1, . . . ,m such thatF

(n)i −→n→∞ Fi, in L

p, and DF (n)i −→n→∞ DFi, in Lp(Ω,F, P ;H). The claim follows.

Taking the gradient of a function lowers the corresponding degree with respect to theWiener chaos in a similar fashion as taking the derivative of a polynomial lowers thedegree by one. The following two propositions describe this relation in detail.

Proposition 2.5.8. Let F ∈ L2(Ω,F, P ) with F =∑∞i=0 JnF corresponding to the

Wiener chaos decomposition. Then F ∈ D1,2 if and only if

E[‖DF‖2H

]=

∞∑n=1

n‖JnF‖22


Proof: First let F = In(fn) where

fn(t1, . . . , tn) =

k∑i1,...,in

ai1...in1Ai1×···×Ain (t1, . . . , tn).

Then

F = In(fn) =

k∑i1,...,in=1

ai1...inW (Ai1) · . . . ·W (Ain)

and F ∈ Z. We get

DtF =

n∑j=1

k∑i1,...,in=1

ai1...inW (Ai1) · . . . · 1Aij (t) · . . . ·W (Ain) = nIn−1(f(·, t))

by symmetry. The claim follows from the closedness of D1,2.

Suppose that F is given. How does one find the representation of F ? The followingproposition provides a tool to achieve this.

Proposition 2.5.10. If F ∈ D∞,2 and

F = E[F ] +

∞∑n=1

In(fn)

then

fk =1

k!E[DkF ], K ∈ N.

Proof: Applying Proposition 2.5.9 k times yields

Dkt1,...,tkF =

∞∑n=k

n(n− 1) . . . (n− k + 1)In−k(fn(t1, . . . , tk; ·))

and thus

1

k!Dkt1,...,tkF = I0(fk(t1, . . . , tk)) +

∞∑n=1

In(gn) = fk(t1, . . . , tk) +

∞∑n=1

In(gn)

where gn is the representing function of the chaos expansion of 1k!Dkt1,...,tk

F . It followsthat

1

k!E[Dkt1,...,tkF ] = fk(t1, . . . , tk).

The following example illustrates the usage of the above proposition.

2.6 Divergence operator 15

Example 2.2Let Wt, t ∈ [0, 1] be a one dimensional Wiener process. The Wiener chaos expansion ofF = W 2aW1, 0 ≤ a ≤ 1 is then

W 2aW1 = 2aWa + aW1 + 4

a∫t3=0

t3∫t2=0

t2∫t1=0

dWt1dWt2dWt3

+ 2

1∫t3=0

t3∫t2=0

t2∫t1=0

1[0,a](t1 ∨ t2) dWt1dWt2dWt3 .

We note that W 2aW1 = W (1[0,a])2W (1[0,1]) and use Proposition 2.5.10 to obtain

f0 = E[W2aW1] = 0,

f1(t1) = E[Dt1W2aW1] = E[DW

2aW1]t1

= E[2W (1[0,a])W (1[0,1])1[0,a](t1) +W (1[0,a])21[0,1](t1)]

= 2E[WaW11[0,a](t1) +W2a1[0,1](t1)] = 2a1[0,a](t1) + a,

f2(t1, t2) =1

2E[D2t1,t2W

2aW1] = . . .

= E[W1]1[0,a](t1 ∨ t2) + E[Wa](1[0,1](t1) + 1[0,a](t2)

)= 0,

where the . . . signifies using the elementary operations as for calculation of f1. In thesame manner we get

f3(t1, t2, t3) =1

3!E[Dt1(Dt2(Dt3W

2aW1))] = . . .

=1

3

(1[0,a](t2 ∨ t3) + 1[0,a](t1 ∨ t2) + 1[0,a](t1 ∨ t3)

).

We recall

F =

1∫0

f1(t1) dWt1 + 3!

1∫t3=0

t3∫t2=0

t2∫t1=0

f3(t1, t2, t3) dWt1dWt2dWt3

and the statement follows.

2.6 Divergence operator

We let Dom δ be the set of all u ∈ L2(Ω,F, P ;H) such that there exists c(u) > 0 with

|E[〈DF, u〉H]| ≤ c‖F‖L2 (2.4)


for all F ∈ D1,2. Then E[〈DF, u〉H] is a bounded linear functional from D1,2 to R. Nowsince D1,2 is dense in L2(Ω,F, P ) Riesz representation theorem states that there exists aunique representing element δ(u), bounded in L2(Ω,F, P ). That is

E[〈DF, u〉H] = E[Fδ(u)], ∀F ∈ D1,2. (2.5)

In other words δ : L2(Ω,F, P ;H)→ L2 and δ is the adjoint operator to D. We call δ thedivergence operator.

Proposition 2.6.1. The divergence operator (δ,Dom δ) has the following properties (i)

E[δ(u)] = 0 ∀u ∈ Dom δ.

(ii) Suppose u ∈ ZH :={v =

∑nj=1 Fjhj : Fj ∈ Z, hj ∈ H

}, then u ∈ Dom δ and

δ(u) =n∑j=1

FjW (hj)−n∑j=1

〈DFj , hj〉H.

(iii) The operator (δ,Dom δ) is closed, i.e. if un ∈ Dom δ, n = 1, 2, . . . , un → u inL2(Ω,F, P ;H), and δ(un)→ G in L2, then u ∈ Dom δ and δ(u) = G.

Proof: (i) This follows from putting F ≡ 1 in (2.5).(ii) Using Lemma 2.5.5 we have

|E[〈DF, u〉H]| =

∣∣∣∣∣∣E n∑j=1

Fj〈DF, hj〉H

∣∣∣∣∣∣=

∣∣∣∣∣∣EF

n∑j=1

−〈DFj , hj〉H +n∑j=1

FjW (hj)

∣∣∣∣∣∣ ≤ c(u)‖F‖L2so u ∈ Dom δ. We also get

E[〈DF, u〉H] = E

F n∑j=1

−〈DFj , hj〉H +n∑j=1

FjW (hj)

= E[Fδ(u)]and the statement follows.(iii) Let un → u in L2(Ω,F, P )). Then for any F ∈ D1,2 we have

E[Fδ(un)] = E[〈DF, un〉H] −→n→∞

E[〈DF, u〉H] = E[Fδ(u)].

Since F is arbitrary it follows that G = δ(u), and also u ∈ Dom δ since (2.4) is satisfiedwith c(u) = ‖G‖L2 .

The following formulas state the relation between divergence δ and gradient D.

2.6 Divergence operator 17

Proposition 2.6.2. Let u, v ∈ ZH, F ∈ Z and h ∈ H. Let {ei : i ∈ N} be an ON-basisin H. Then

a) Dh(δ(u)) = 〈u, h〉H + δ(Dh(u)).

b) E[δ(u)δ(v)] = E[〈u, v〉H] + E

∞∑i,j=1

Dei〈u, ej〉HDej 〈v, ei〉H

.c) δ(Fu) = Fδ(u)− 〈DF, u〉H.

Proof: a) Let u =∑nj=1 Fjhj where FjZ, hj ∈ H, j = 1, . . . , n. Now

Dh(δ(u)) = 〈D(δ(u)), h〉H =〈D

Proposition 2.6.1(ii)︷︸︸︷( n∑j=1

FjW (hj)−n∑j=1

〈DFj , hj〉H), h

〉H

=

〈n∑j=1

FjDW (hj) +

n∑j=1

DFjW (hj)−n∑j=1

D(〈DFj , hj〉H

), h

〉H

=

n∑j=1

Fj〈h, hj〉H +n∑j=1

DhFjW (hj)−n∑j=1

Dh

(〈DFj , hj〉H

)= 〈u, h〉H +

n∑j=1

(DhFj)W (hj)−n∑j=1

〈D(DhFj), hj〉H)

= 〈u, h〉H + δ(Dh(u))

where we in the last line once again used Proposition 2.6.1(ii).

b) Using a) we obtain

E[δ(u)δ(v)] = E[〈u,Dδ(v)〉H] = E

[ ∞∑i=1

〈u, ei〉HDeiδ(v)

]

= E

[ ∞∑i=1

〈u, ei〉H(〈v, ei〉H + δ

(Dei(v)

))]

= E [〈u, v〉H] +∞∑i=1

E[〈u, ei〉Hδ

(〈Dv, ei〉H)

]= E [〈u, v〉H] + E

∞∑i=1

〈 ∞∑j=1

Dej 〈u, ei〉Hej , Dei

( ∞∑k=1

〈v, ek〉ek

)〉H

= E [〈u, v〉H] + E

∞∑i,j=1

Dej 〈u, ei〉H ·Dei〈v, ej〉H

.


c) For G ∈ Z we have

E[Gδ(Fu)] = E[〈DG,Fu〉H = E[〈FDG, u〉H] = E[〈u,D(FG)−GDF 〉H]= E[(δ(u)F − 〈u,DF 〉H)G].

Since G was arbitrary it follows δ(Fu) = Fδ(u)− 〈DF, u〉H.

We state the connection between the Wiener chaos expansion and the divergence op-erator. Recall that for u ∈ L2([0, 1]× Ω), u has a Wiener chaos expansion of form

u(t) =

∞∑n=0

In(fn(·, t)),

where fn ∈ L2([0, 1]n) and fn is symmetric in the n first variables.

Proposition 2.6.3. We have u ∈ Dom δ if and only if the series∞∑n=0

In+1(f̃n)

converges in L2(Ω,F, P ). In this case we have

δ(u) =

∞∑n=0

In+1(f̃n).

Proof: Suppose G = In(g) where g ∈ L2s([0, 1]n). Now using Proposition 2.5.9 we get

E[〈u,DG〉H] = E[〈u, nIn−1(g(·, t))〉H] = E[〈In−1(fn−1(·, t)), nIn−1(g(·, t))〉H]

=

∫[0,1]

E[In−1(fn−1(·, t))nIn−1(g(·, t))] dt

= n(n− 1)!∫

[0,1]

〈fn−1, g〉L2([0,1]n−1) dt

= n!〈fn−1, g〉L2([0,1]n = n!〈f̃n−1, g〉L2([0,1]n)= E[In(f̃n−1)In(g)] = E[In(f̃n−1)G]

where the third and last lines come from the isometry between L2s([0, 1])n and Hn.

One can see from this that δ is an integral. δ(u) is called the Skorokhod stochasticintegral of the process u and if the process is adapted then the Skorokhod integral willcoincide with the Itô integral, i.e.

δ(u) =

1∫0

ut dWt.

2.7 The generator of the Ornstein-Uhlenbeck semigroup 19

2.7 The generator of the Ornstein-Uhlenbecksemigroup

The Ornstein-Uhlenbeck semigroup is related to the solution of some stochastic differen-tial equations. It is closely related to the gradient and divergence. The generator of thissemigroup is denoted L. In this presentation we start with the following definition.

Definition 2.7.1. For F ∈ L2(Ω,F, P ) define

LF := −∞∑n=0

nJnF

where Jn denotes the projection to the nth Wiener chaos. We define Dom L as

Dom L :=

{F ∈ L2 :

∞∑n=0

n2‖JnF‖22


Proposition 2.7.3. We have Z ∈ Dom δ and for F ∈ Z withF = f(W (h1), . . . ,W (hn)), f ∈ C∞p (Rn) we have

LF =

n∑i,j=1

∂2

∂xi∂xjf(W (h1), . . . ,W (hn))〈hi, hj〉H

−n∑i

∂

∂xif(W (h1), . . . ,W (hn))W (hi).

Proof: Since F ∈ D1,2 and DF =∑ni=1

∂∂xi

f(W (h1), . . . ,W (hn))hi we have thatDF ∈ ZH. We recall that for

u :=

n∑j=1

Fjhj , Fj ∈ D1,2, hj ∈ H,

we have

δ(u) =

n∑j=1

FjW (hj)−n∑j=1

〈DFj , hj〉H .

We now get

δDF =

n∑i=1

∂

∂xif(W (h1), . . . ,W (hn))W (hi)

−n∑j=1

〈D

(∂

∂xjf(W (h1), . . . ,W (hn))

), hj

〉H

=

n∑i=1

∂

∂xif ·W (hi)−

n∑j=1

〈n∑i=1

∂

∂xif · hi, hj〉H

=

n∑i=1

∂

∂xif ·W (hi)−

n∑i,j=1

∂

∂xi∂xjf · 〈hi, hj〉H

and the claim follows.

We see that the Malliavin calculus provides the necessary tools for doing analysis onthe Wiener space, the most prominent tool being the integration by parts formula.

3Chapter 3-Dirichlet form theory

DIRICHLET forms are a special type of bilinear forms. The theory of Dirichlet formswas first introduced in the works by Beurling and Deny in 1958 and 1959. There is alink between Dirichlet forms and Markov processes. In this chapter the general definitionsare introduced and the links between semigroup, resolvent, generator, and form are stated.

3.1 Introduction

This chapter follows the structure in the book by Z.-M. Ma and M. Röckner [5] and thetheory can be found there in a more general form. In this section we will let H be a Hilbertspace with corresponding norm ‖ · ‖H and inner product 〈·, ·〉H.

3.2 Resolvents

The resolvent set of a linear operator L is a set of complex numbers λ for which theoperator L− λI is in some sense well-behaved.

Definition 3.2.1. Let L be a linear operator on H. The resolvent set ρ(L) of L is definedto be the set of all α ∈ R such that (α−L) : D(L)→ H is one-to-one and for its inverse(α− L)−1 we have

(i) D((α− L)−1) = H,(ii) (α− L)−1 is continuous on H.

We call {(α−L)−1 : α ∈ ρ(L)} the resolvent of L and σ(L) := R \ ρ(L) the spectrumof L.

We see that all eigenvalues are part of the spectrum but the spectrum can have otherpoints as well. For α ∈ ρ(L) we let Ga := (α− L)−1.

21

22 3 Chapter 3-Dirichlet form theory

Definition 3.2.2. A family (Gα)α>0) of linear operators on H is called a continuouscontraction resolvent on H if:

(i) limα→∞

αGαu = u for all u ∈ H, (continuity)

(ii) Ga is a contraction on H for all α > 0,(iii) Gα −Gβ = (β − α)GαGβ for all α, β > 0. (resolvent equation)

3.3 Semi-groups

In probability theory, semigroups are associated with the transition probabilities of Markovprocesses.

Definition 3.3.1. A family (Tt)t>0 of linear operators on H with D(Tt) = H for allt > 0 is called a continuous contraction semigroup on H if

(i) limt→0

Ttu = u for all u ∈ H, (continuity)

(ii) Tt is a contraction on H for all t > 0,(iii) TsTt = Ts+t for all s, t > 0. (semigroup property)

It turns out that semigroups are connected to resolvents as will be shown later.

Definition 3.3.2. The infinitesimal generator of a continuous semigroup (Tt)t>0 is de-fined as the operator (A, D(A)) on H satisfying

D(A) = {u ∈ H : limt↘0

Ttu− ut

exists},

Au = limt↘0

Ttu− ut

, u ∈ D(A).

Given a semigroup and thus also its generator, it is possible to, in a natural way,associate it with a resolvent. The following proposition shows this relation.

Proposition 3.3.3. Let (Tt)t>0 be a continuous contraction semigroup on H with gen-erator A. Then A is densely defined, (0,∞) ⊂ ρ(A) and if Gα := (α −A)−1, α > 0then

Gαu =

∞∫0

e−αsTsu ds, u ∈ H, α > 0.

We also have that (Gα)α>0 is a continuous contraction resolvent.

Proof: The proof of this proposition will be omitted but can be found in [5].

We see that there is a connection between semigroups, resolvents, and the generatorA. For an arbitrary linear operator L one might wonder if it is the generator of a semi-group or not. The answer to this quesion is answered by the Hille–Yosida theorem, whichcharacterizes the generators for contraction semigroups. The theorem is named after themathematicians Einar Hille and Kosaku Yoside who independently discovered the resultin 1948.

3.4 Bilinear forms 23

Theorem 3.3.4. (Hille–Yosida) A densely defined linear operator A on H is the genera-tor of a continuous contraction semigroup if any only if

(i) (0,∞) ⊂ ρ(A),(ii) ‖α(α−A)−1‖ ≤ 1 for all α > 0.

The continuous contraction semigroup is uniquely determined by A and A is closed.

Proof: The proof of the Hille–Yosida theorem is quite lengthy and will therefore be omit-ted. The full proof can be found in [5].

3.4 Bilinear forms

In this section we’ll look into the relationship between the resolvent, semigroup and bi-linear form. We will assume that E is positive definite throughout the section.

Definition 3.4.1. The symmetric part Ẽ of a bilinear form E is defined by

Ẽ(u, v) =1

2(E(u, v) + E(v, u)).

Definition 3.4.2. We define

Eα(u, v) = E(u, v) + α〈u, v〉H, u, v ∈ D(E).

Following this definition we can also define the corresponding norm.

Definition 3.4.3. Given a bilinear form E on a Hilbert space H we define the E1-normby

‖ · ‖E1 :=(〈·, ·〉H + E(·, ·)

)1/2.

Definition 3.4.4. (E, D(E)) satisfies the weak sector condition if there exists a constantK > 0 such that

|E1(u, v)| ≤ KE1(u, u)1/2E1(v, v)1/2,∀u, v ∈ D(E).

Definition 3.4.5. A bilinear form (E, D(E)) on H is called coercive if D(E) is a denselinear subspace of H and E : H×H→ R such that

(i) Its symmetric part (Ẽ, D(E)) is a symmetric closed form on H.(ii) (E, D(E)) satisfies the weak sector condition.

A property of a coercive form is that it grows rapidly as the argument gets bigger.

Theorem 3.4.6. Let (E, D(E)) be a coercive closed form on H. Then there exist uniquecontinuous contraction resolvents (Gα)α>0, (G̃α)α>0 on H such that

Gα(H), G̃α(H) ⊂ D(E),Eα(Gαf, u) = 〈f, u〉H = Eα(u, G̃αf) for all f ∈ H, u ∈ D(E), α > 0.

G̃α is the adjoint of Gα for all α > 0 and we have

〈Gαf, g〉H = 〈f, G̃αg〉 for all f, g ∈ H.


Proof: The proof of this theorem will be omitted. As in the other cases the full proof canbe found in [5].

A consequence of this is the following relation between the bilinear form and thecorresponding generator to the resolvent.

Corollary 3.4.7. Let (E, D(E)) and (Gα)α>0 be as above and let (A, D(A)) be thegenerator of (Gα)α>0 i.e. the unique operator on H such that (0,∞) ∈ ρ(A) and Gα =(α−A)−1 for all α > 0. Then

D(A) ⊂ D(E),E(u, v) = 〈−Au, v〉H for all u ∈ D(A), v ∈ D(E).

Proof: Let α > 0 and u ∈ D(A). Now u ∈ Gα(H) ⊂ D(E) and we get for all v ∈ D(E)

E(u, v) = Eα(GαG−1α u, v)− α〈u, v〉H = 〈G−1α u, v〉H − α〈u, v〉H

=〈(G−1α − α)u, v

〉H = 〈−Au, v〉H.

Theorem 3.4.8. Let (Gα)α>0 be a continuous contraction resolvent on H with corre-sponding generator A. Define

E(u, v) := 〈−Au, v〉H, u, v ∈ D(A).

Now (Gα)α>0 and (E, D(E)) have the relation as in the above theorem.

These theorems state the relationship between the resolvent to a linear operator andits corresponding bilinear form.

3.5 Dirichlet forms

The theory of Dirichlet forms is used in the area of Markov process theory. Dirichlet formsare connected to potential theory and energy methods in contrast to partial differentialequation theory which are the usual tools used in diffusion theory.

Definition 3.5.1. A coersive closed form (E, D(E)) on L2(E,m) is called a Dirichletform if for all u ∈ D(E) it holds

(i) u+ ∧ 1 ∈ D(E),(ii) E(u+ u+ ∧ 1, u− u+ ∧ 1) ≥ 0,(iii) E(u− u+ ∧ 1, u+ u+ ∧ 1) ≥ 0.

Remark 3.5.2. If E is symmetric a simple calculation shows that conditions (ii) and (iii)above are equivalent to

E(u+ ∧ 1, u+ ∧ 1) ≤ E(u, u).

3.5 Dirichlet forms 25

Definition 3.5.3. Let H = L2(E,m). A bilinear form E on L2(E,m) is regular ifD(E) ∩ C0(E) is dense in D(E) w.r.t. E1-norm and dense in C0(E) w.r.t. the uniformnorm. Here C0(E) denotes all continuous functions on E with compact support.

We recall some basic definitions from the theory of stochastic processes.

Definition 3.5.4. We say that a function is cadlag if it is right continuous with left limits,i.e.

(i) f(x−) := limt↗x

f(t) exists,

(ii) f(x+) := limt↘x

f(t) exists and is equal to f(x).

The corresponding term for left continuous functions is caglad.

Definition 3.5.5. Two stochastic processes X and Y with a common index set T arecalled versions of one another if

t ∈ T, P ({ω : X(t, ω) = Y (t, ω)}) = 1.

Such processes are also said to be stochastically equivalent.

If in addition the process Xt is left or right continuous then for a version Yt we haveXt = Yt almost surely.

It turns out that we only need to study the form on a dense subset of D(E) and herethe concept of nests appear.

Definition 3.5.6. (i) An increasing sequence (Fk)k∈N of closed subsets of E is calledan E-nest if ∪k≥1D(E)Fk is dense in D(E) w.r.t. ‖ · ‖Ẽ1 , where D(E)Fk denotes {u ∈D(E) : u = 0 m-a.e. on E \ Fk}.(ii) A subset N ⊂ E is called E-exceptional if N ⊂ ∩k≥0F ck for some E-nest (Fk)k∈N.(iii) We say that a property holds E-quasi-everywhere if it holds everywhere outsidesome E-exceptional set.

We can relate the definition of quasi-continuity to a similar notion on the E-nest.

Definition 3.5.7. An E-quasi-everywhere defined function f is called E-quasi-continuousif there exists an E-nest (Fk)k∈N such that f is continuous on (Fk)k∈N.

It is now possible to define what we mean with a quasi-regular Dirichlet form.

Definition 3.5.8. A Dirichlet form (E, D(E)) on L2(E,m) is called quasi-regular if:

(i) There exists an E-nest (Fk)k∈N consisting of compact sets,

(ii) There exists an Ẽ1/21 -dense subset of D(E) whose elements haveE-quasi-continuous m-versions,

(iii) There exist un ∈ D(E), n ∈ N, having E-quasi-continuous m-versions ũn,n ∈ N, and an E-exeptional set N ⊂ E such that {ũn : n ∈ N} separates thepoints of E \N.


In infinite dimension the quasi-regularity of the Dirichlet form is enough to be able toassociate it with a special kind of process. The following definitions serve to clarify whatwe mean.

Definition 3.5.9. A process M with state space E is called µ-tight if there exists anincreasing sequence (Kn)n∈N of compact sets in E such that

Pµ( limn→∞

inf{t > 0 : Mt ∈ E \Kn} 0 there exists a compact set K suchthat Pµ(Mt ∈ K) > 1− ε.

Definition 3.5.10. A cadlag Markov process M with state space E and transition semi-group (pt)t>0 is said to be properly associated with (E, D(E)) and its correspondingsemigroup (Tt)t>0 if ptf is an E-quasi continuous µ-version of Ttf for all t > 0 and allbounded f ∈ L2(E;µ).

The following theorem is the connection between the theory of forms and processtheory.

Theorem 3.5.11. Let (E, D(E)) be a quasi-regular Dirichlet form on L2(E,µ). Thenthere exists a pair (M,M̂) of µ-tight special standard processes which is properly associ-ated with (E, D(E)).

Proof: The proof is lengthy and will therefore be omitted. The reader may find it in[5].

Remark 3.5.12. It is known that if E is locally compact then there exists an associatedprocess to every regular Dirichlet form on L2(E,µ). See [3].

The Cameron–Martin formula tells us how the Wiener measure changes under a trans-lation. It was discovered by Robert Cameron and William Martin in 1944.

Theorem 3.5.13. Let h ∈ H, where

H = {f ∈ C([0, 1]) : f(0) = 0, f absolutely continuous,1∫

0

f ′(u)2 du

3.5 Dirichlet forms 27

Proof: Let f, h ∈ H and α := 〈f, h〉H/〈h, h〉H. Then Wh and Wf−αh are independentsince they are Gaussian and

E[WhWf−αh] = 〈h, f − αh〉H = 〈h, f〉H − 〈f, h〉H = 0.

We have∫exp {i〈f, z〉H} exp

{Wh(z)−

1

2‖h‖2H

}µ(dz)

= exp

{−1

2‖h‖2H

}∫exp {i〈f, z〉H + 〈h, z〉H} dµ

= exp

{−1

2‖h‖2H

}∫exp {i〈f − αh, z〉H + 〈(1 + iα)h, z〉H} dµ

= exp

{−1

2‖h‖2H

}∫exp {iWf−αh + (1 + iα)Wh} dµ

= exp

{−1

2‖h‖2H

}∫exp{iWf−αh} dµ

∫exp{i(−i+ α)Wh} dµ

= exp

{−1

2‖h‖2H

}exp{−1

2‖f − αh‖2H} exp{−

1

2(−i+ α)2‖h‖2H}

= exp

{−1

2‖h‖2H −

1

2‖f − αh‖2H +

1

2‖h‖2H + iα‖h‖2H −

1

2α2‖h‖2H

}= exp

{−1

2〈f, f〉H + i〈f, h〉H

}=

∫exp {i〈f, z〉H} (µ ◦ T−1h )(dz).

Keeping all these definitions and theorems in mind we are now well equipped to studythe Dirichlet form in the next chapter.

4Chapter 4-Main results

THE paper included here is a joint work with Jörg-Uwe Löbus and is the result of astudy at the department of Mathematical Statistics at Linköping University during2012. This version of the paper contains slightly more details than the version availableat ArXiv and the version submitted for publication. The aim of the paper is to take resultsfrom the finite dimensional theory into the infinite dimensional case. This is done via themeans of a coordinate representation.

4.1 Earlier work

In the paper Construction of diffusions on path and loop spaces of compact Riemannianmanifolds [2] the idea of Dirichlet form theory is used to create a diffusion process. Moreprecisely the idea is that every quasi-regular Dirichlet form has an associated diffusionprocess. In [2] a form defined on a compact Riemannian manifold with these propertiesis presented.The paper A class of processes on the path space over a compact riemannian manifoldwith unbounded diffusion [4] expands on the same idea. Here the form is∫

〈DF,DAG〉H dν (4.1)

where the operator A is defined by

AΦ(γ) :=

∞∑i=1

λi〈Si,Φ(γ)〉HSi,Φ ∈ D(A),

and the diffusion coefficients λ are assumed to be bounded from below and possiblyunbounded. It is shown that under the assumption λn ≤ cn1−ε, the form is a quasi-regular Dirichlet form and thus has an associated process.

29

30 4 Chapter 4-Main results

The paper presented in this thesis is similar to the paper Quasi-regular Dirichlet forms onRiemannian path and loop spaces [11]. Both these papers investigate infinite dimensionaldiffusion processes where the diffusion is related to possibly unbounded operators. Bothpapers refer to the ideas of [4]. Let us summarize the differences between this paper and[11]. The main difference is that the theory in [11] is presented in a geometric setting, i.e.on a manifold while here we study the flat case. Even though the geometric frameworkin [11] is more general, our coordinatewise non-geometric setting makes it easier to focuson the phenomenon we are interested in to investigate, namely pointwise unboundednessof diffusion. The forms studied in these two papers are of type (4.1). The form in [11]differs from the one presented here in the sense that the operatorA in this paper is definedpointwise, constant over the trajectory space, while in [11] it can vary, i.e. A = A(γ). Theoperator in [11] is defined by letting (A,D(A)) be a densely defined self-adjoint operatoron L2(W0 → H;µ) such that

AfΦ = fAΦ for f ∈ L∞(µ) and Φ ∈ L2(W0 → H;µ),and A ≥ εI for some ε > 0.

(A0)

Here W0 is a manifold and I denotes the identity operator. They define

EA(F,G) =

∫W0

〈A1/2DF,A1/2DG〉H dµ, F, g ∈ Y,

whereY := {F ∈ FC∞b : DF ∈ D(A1/2)},

and FC∞b denotes the set of cylindrical functions {F (γ) = f(γ(s1), . . . , γ(sk)) : sj ∈[0, 1], f ∈ C∞b }. In addition the assumption

‖A1/2Φt,v‖2L2(W0→H;µ) ≤ C‖v‖2L∞(W0→Rd;µ), t ∈ [0, T ], v ∈ L

∞(W0 → Rd;µ),(A1)

is made. In our situation with A constant on the trajectory space and with non-decreasingeigenvalues, the conditions (A0) and (A1) are satisfied. In [11] the condition

d∑j=1

µ(λi) +

∞∑m=0

d∑j=1

max1≤k≤2m

µ(λd(2m+k−1)+j)2−m

4.2 Introduction 31

4.2 Introduction

This paper is concerned with Dirichlet forms of type

E(F,G) =

∫〈DF,ADG〉H ϕdν =

∫ 〈DF,

∞∑i=1

λi 〈Si, DG〉H Si

〉H

ϕdν, (4.2)

where the diffusion operator A is in general unbounded, cf. [4]. We are interested inweight functions ϕ of the form

ϕ(γ) = exp

1∫

0

〈b(γs), dγs〉Rd −1

2

1∫0

‖b(γs)‖2 ds

.This choice of weight functions is motivated by the papers [11, 12] by F.-Y. Wang andB. Wu. In this way it is possible to relate and compare the results of the present paperto the findings there. We present our ideas in terms of infinite dimensional processeson the classical Wiener space using a coordinate representation. This makes the subjectcomprehensible, in particular, to readers familiar with the finite dimensional theory. Theform is studied on the set of smooth cylindrical functions

F,G ∈ Y = {F (γ) = f(γ(s1), . . . , γ(sk)) : sj is a dyadic point},where γ is a Wiener trajectory. We do this as well on the more common set of cylindricalfunctions

F,G ∈ Z = {F (γ) = f(γ(s1), . . . , γ(sk)) : sj ∈ [0, 1]}.Well-definiteness of E on Y is a consequence of the fact that the sum in (4.2) is finite.Well-definiteness of E on Z requires the convergence of the sum in (4.2). These twodifferent initial situations result in possibly different closures of (E, Y ) and (E, Z) onL2(ϕν). Using the coordinate representation in (4.2) we give conditions for closability.The requirement of ϕ−1 ∈ L1(ν) would be sufficient, for example, for closability in theclassical case with bounded cylindrical functions and λ1 = λ2 = . . . = 1 on L2(ν),see [5]. However, the paper investigates forms of the structure (4.2) with an in generalunbounded diffusion operator. We give necessary and sufficient conditions on the increaseof λ1, λ2, . . . that guarantee closability of (E, Y ) and (E, Z) on L2(ϕν) in terms of theSchauder functions Si, i ∈ N, the coordinate functions in H. Locality, Dirichlet property,and quasi-regularity of the closure (E, Z) on L2(ϕν) is then obtained by using methodsof [1, 2, 4, 5, 9]. We are also interested in characterizing the associated process in termsof their local first and second moments of the form

limt→0

1

t

∫(Si(γ)− Si(τ)) Pτ (Xt ∈ dγ), (4.3)

andlimt→0

1

t

∫(Si(γ)− Si(τ))2 Pτ (Xt ∈ dγ), (4.4)

where Si are certain linear functions on the trajectory space. We represent these localmoments in terms of the sequence λ1, λ2, . . . and the weight function ϕ. Correspondingto the closability condition of (E, Z) on L2(ϕν), we derive a necessary and sufficientcondition for the limits (4.3) and (4.4) to exist. This way lets us obtain compatibility withthe classical Kolmogorov characterization of finite dimensional diffusion processes.


4.3 Formal definitions

We study the form on the space L2(ϕν) ≡ L2(Ω, ϕν) where Ω := C0([0, 1];Rd) :={f ∈ C([0, 1];Rd, f(0) = 0}, ν is the Wiener measure on Ω and ϕ is a density functionspecified below. As stated earlier, the form is given by

E(F,G) =

∫〈DF,ADG〉H ϕdν, F,G ∈ D(E), (4.5)

where H is the Cameron-Martin space, i.e., the space of all absolutely continuous Rd-valued functions f on [0, 1], with f(0) = 0 and equipped with inner product

〈ϕ,ψ〉H :=∫

[0,1]

〈ϕ′(x), ψ′(x)〉Rd dx.

Motivated by [11], we suppose in sections 4.7 and 4.8 that ϕ : Ω→ [0,∞] has the form

ϕ(γ) = exp

1∫

0

〈b(γs), dγs〉Rd −1

2

1∫0

‖b(γs)‖2Rd ds

(4.6)where b : Rd → Rd is the gradient of a function f ∈ C2b (Rd;R). Then by Itô’s formula,ϕ defined by (4.6) is both bounded from below and from above, on Ω. We define the setof all cylindrical functions

Z :={F (γ) = f (γ(s1), . . . , γ(sk)) , γ ∈ Ω :

0 < s1 < · · · < sk = 1, f ∈ C∞p (Rdk), k ∈ N

}where C∞p denotes smooth functions with polynomial growth. We also define

Y :={F (γ) = f (γ(s1), . . . , γ(sk)) , γ ∈ Ω :

F ∈ Z, s1, . . . , sk ∈{l

2n : l ∈ {1, . . . , 2n}}, n ∈ N

}.

For F ∈ Z and γ ∈ Ω the gradient operator D is defined by

DsF (γ) =

k∑i=1

(si ∧ s)(∇sif)(γ), s ∈ [0, 1], (4.7)

where (∇sif)(γ) = (∇sif)(γ(s1), . . . , γ(sk)) denotes the gradient of the function frelative to the ith variable while holding the other variables fixed. We let (ej)j=1,...,ddenote the standard basis in Rd and

H1(t) = 1, t ∈ [0, 1],

H2m+k(t) =

2m/2 if t ∈

[k−12m ,

2k−12m+1

)−2m/2 if t ∈

[2k−12m+1 ,

k2m

)k = 1, . . . , 2m, m = 0, 1, . . . ,

0 otherwise

4.4 Definition of the form 33

denote the system of the Haar functions on [0, 1]. We also define

gd(r−1)+j := Hr · ej , r ∈ N, j ∈ {1, . . . , d}, (4.8)

and

Sn(s) :=

s∫0

gn(u) du, s ∈ [0, 1], n ∈ N.

4.4 Definition of the form

We use the following definitions from [4]. We choose a non-decreasing sequence ofpositive numbers λ1, λ2, . . . .

D(A) :=

{Φ ∈ L2(Ω→ H, ν) :

∫ ∞∑i=1

λ2i 〈Si,Φ〉2H dν


where f (n)j ∈ L2(T j+1) where for j ≥ 1, f(n)j is symmetric in the first j variables. Thus

δ(xn) =

∞∑j=0

Ij+1(f̃(n)j ).

We can restrict ourselves to φ ∈ Hm+1 such that φ ∈ Z. By hypothesis we have

(m+ 1)!〈g, f̃ (n)m

〉L2(Tm+1)

=

∫Im+1(g)Im+1(f

(n)m ) dν

=

∫φδ(xn) dν → 0, as n→∞.

Since g ∈ L2(Tm+1) is symmetric we have〈g, f (n)m

〉L2(Tm+1)

→ 0, as n→∞.

For all symmetric h ∈ L2(Tm,H) we obtain

〈h, f (n)m (·, t)〉L2(Tm;H) → 0, as n→∞,

where the letter t indicates the variable for the function in H. Using the fact that the mthWiener chaos is isometric to the space of symmetric functions L2s(T

m) we get for anyψ ∈ L2(ν;H) such that ψ(·, t) ∈ Hm for a.e. t ∈ [0, 1], ψ = Im(h), we have〈

Im(h), Im(f (n)m (·, t)

)〉L2(ν;H)

→ 0, as n→∞.

Since Im(f

(n)m (·, t)

)is the projection of xn to the mth Wiener chaos we have

〈xn, φ̃〉L2(ν;H) −→n→∞

0,

for all finite linear combinations φ̃ = α1ψ1 + . . .+ αkψk, where ψi ∈ L2(ν;H), ψi(·, t)is from the ith Wiener chaos for a.e. t ∈ [0, 1], and i ∈ {1, . . . , k}. Since ϕ is boundedthis implies

〈xn, φ̃〉L2(ϕν;H) −→n→∞

0.

Since also ‖xn‖L2(ϕν;H) is bounded we have

〈xn, φ̃〉L2(ϕν;H) −→n→∞

0, for all φ̃ ∈ L2(ϕν;H).

This contradicts xn → x in L2(ϕν;H) and the statement follows.

Lemma 4.5.2. Iff

ϕ∈ L1(ν), ∀f ∈ Z, (4.12)

then { 1ϕDψ : ψ ∈ Z} is a dense subset of L2(ϕν;H).

4.5 Closability 35

Proof: We assume the contrary. Then we can find x ∈ L2(ϕν;H) and cylindrical func-tions xn, n ∈ N, with values in H, such that x 6= 0 and xn −→

n→∞x in L2(ϕν;H) and∫

1

ϕ〈Dψ, x〉Hϕdν = 0

for all ψ ∈ Z. It follows that

0 =

∫1

ϕ〈Dψ, x〉H ϕdν = lim

n→∞

∫〈Dψ, xn〉H dν = lim

n→∞

∫ψδ(xn) dν

for all ψ ∈ Z. From this we get x = 0 by Lemma 4.5.1, and we have a contradiction.

Lemma 4.5.3. If 0 < ϕ ≤ c for some c ∈ R+, and (4.12) holds, then (D , Z) defined by

D(f, g) :=1

2

∫〈Df,Dg〉H ϕdν, f, g ∈ Z,

is closable on L2(ϕν). Let L denote the generator of D and note that L is the Ornstein-Uhlenbeck operator.

Proof: Let un ∈ Z, ψ ∈ Z and

un −→n→∞

0 in L2(ϕν), Dun −→n→∞

f in L2(ϕν;H).

Then

1

2

∫ 〈Dun,

1

ϕDψ

〉Hϕdν =

1

2

∫〈Dun, Dψ〉H dν

=

∫un(−Lψ) dν

=

∫un

1

ϕ(−Lψ)ϕdν.

By assumption we have 1/ϕ · (−Lψ)2 ∈ L1(ν), i.e. 1/ϕ · (−Lψ) ∈ L2(ϕν). Thus∫un

1

ϕ(−Lψ)ϕdν −→

n→∞0.

Since Lemma 4.5.2 showed that { 1ϕDψ : ψ ∈ Z} is a dense subset of L2(ϕν;H), it now

follows that f = 0. Hence the form is closable.

Remark 4.5.4. The closability condition (4.12) would weaken to 1/ϕ ∈ L1(ν) if theinitial definition of D was on all bounded f, g ∈ Z. We get compability with [5], sectionII.2 a).

We also formulate the following lemma that serves an important role in proving andformulating the closability conditions of the form.


Lemma 4.5.5. (a) If sk =∑ri=1 ci · 2−i, ip,j = 2p−1(d+ j− 1) + 1 +

∑p−1q=0 cq2

p−q−1

and j ∈ {1, . . . , d} where c1, . . . , cr−1 ∈ {0, 1}, cr = 1, c0 = cr+1 = 0 and r ≥ 2, then

∞∑i=1

λi〈Si(sk), ej〉2Rd = λjs2k +

r∑p=1

λip,j2p−1

(cp2−p + (−1)cp

r+1∑q=p+1

2−qcq

)2.

(b) The relation

supc1,...,cr−1∈{0,1},

cr=1,cr+1=0,j∈{1,...,d},r≥2

r∑p=1

λip,j2p−1

(cp2−p + (−1)cp

r+1∑q=p+1

2−qcq

)2

4.5 Closability 37

Using c1 = c3 = c5 = · · · = 1, c2 = c4 = c6 = · · · = 0 we get

supc1,...,cr−1∈{0,1},cr=1,cr+1=0,r≥2

r∑p=1

λip,j2p−1

(cp2−p + (−1)cp

r+1∑q=p+1

2−qcq

)2

≥∞∑p=1

λip,j2p−1 (2−p−1)2 = 1

8

∞∑p=1

λip,j2p

≥ 18

∞∑p=1

λd2p−1

2p=

1

16

∞∑p=0

λd2p

2p.

Assuming (4.14) we obtain

supc1,...,cr−1∈{0,1},cr=1,cr+1=0,r≥2

r∑p=1

λip,j2p−1

(cp2−p + (−1)cp

r+1∑q=p+1

2−qcq

)2

≤∞∑p=1

λip,j2p−1 (2−p)2 = 1

2

∞∑p=1

λip,j2p≤ 1

2

∞∑p=1

λd2p

2p.

The statement now follows.

Proposition 4.5.6. Let ϕ satisfy (4.12).(a) The form (E, Y ) is closable in L2(ϕν). Let (E, DY (E)) denote the closure of (E, Y )in L2(ϕν).(b) If

∞∑p=0

λd2p

2p


The operator J is bounded in L2(Ω→ H, ϕν) and it follows

DFn = JA1/2DFn −→

n→∞Jψ in L2(Ω→ H, ϕν).

From Lemma 4.5.3, it is known that (D,Z) is closable on L2(ϕν). It follows thatDFn −→

n→∞0 and thus Jψ = 0. Since λi > 0 and λ

−1/2i > 0, (4.16) gives ψ = 0.

Now A1/2DFn −→n→∞

0 and thus E(Fn, Fn) =∫ 〈

A1/2DFn, A1/2DFn

〉H ϕdν → 0 as

n→∞.(b) We show that (4.15) implies Z ⊂ DY (E). Then Y ⊂ Z ⊂ DY (E). Since (E, Y ) isclosable with closure (E, DY (E)), cf. (a), (E, Z) is then also closable and has (E, DY (E))as its closure i.e. DY (E) = DZ(E).Let xv(p) denote the vth coordinate of p ∈ Rd, v ∈ {1, . . . , d}. Let us demonstrate thatF (γ) = xv(γ(s)) ∈ DY (E) for all v ∈ {1, . . . , d} and all s ∈ [0, 1]. Fix s ∈ [0, 1],v ∈ {1, . . . , d}, and let sk −→

k→∞s where sk is a sequence of dyadic numbers. Now let

Fv,k(γ) := xv(γ(sk)) ∈ Y ⊂ DY (E), k ∈ N.

Using Lemma 4.5.5 we have∞∑i=1

λi〈Si, DF (γ)〉2H =∞∑i=1

λi〈Si(s), ev〉2Rd ≤ supk

∞∑i=1

λi〈Si(sk), ev〉2Rd

≤ λv +1

2

∞∑p=1

λip,v2p≤∞∑p=0

λd2p

2p

4.6 Quasi-regularity 39

Proposition 4.5.7. Let ϕ satisfy (4.12). The form (E, DZ(E)) is a Dirichlet form onL2(ϕν).

Proof: We use Proposition I.4.10 from [5]. It follows that we must show E(1 ∧ F+, 1 ∧F+) ≤ E(F, F ). We know that for F ∈ Y

E(F, F ) =

∞∑i=1

λi

∫〈Si, DF 〉2H ϕdν =

∞∑i=1

λi

∫(∂Si , F )

2 ϕdν

=

∞∑i=1

λi

∫ (d

dt

∣∣∣∣t=0

F (γ + tSi)

)2ϕdν.

Let ξε : R → [−ε, 1 + ε] be non-decreasing such that ξε(t) = t for all t ∈ [0, 1],0 ≤ ξ′ε ≤ 1. It follows that ξε ◦ F → 1 ∧ F+. An application of the chain rule gives

E(ξε ◦ F, ξε ◦ F ) =∫ (

d

dt

∣∣∣∣t=0

ξε ◦ F (γ + tS))2

ϕdν

=

∫ξ′ε(F (γ)) ·

(d

dt

∣∣∣∣t=0

F (γ + tS)

)2ϕdν

≤∫ (

d

dt

∣∣∣∣t=0

1 · F (γ + tS))2

ϕdν

= E(F, F ),

from which we derive E(1 ∧ F+, 1 ∧ F+) ≤ E(F, F ). Thus E is a Dirichlet form.

Proposition 4.5.8. Let ϕ satisfy (4.12). The form (E, DZ(E)) is local.

Proof: This is shown in the same manner as in Proposition 3.4 of [4] using the fact thatν-a.e. implies ϕν-a.e.

4.6 Quasi-regularity

It turns out that the closability condition found in the previous sections is sufficient forthe form to be quasi-regular. Throughout this section we assume (4.12).

Proposition 4.6.1. Suppose (4.15). The form described by the closure of

E(F, F ) =

∞∑i=1

λi

∫〈Si, DF 〉2H ϕdν, F ∈ Z,

in L2(ϕν), is quasi-regular.

Proof: The conditions for closability can be found in Proposition 4.5.6. We follow theprocedure of [4], see also [2, 9].Step 1 : For r ∈ N, l ∈ {0, . . . , 2r−1 − 1}, and k = 2r−1 + l, set sk := (2l + 1)2−r. Let


xv(p) denote the vth coordinate of p ∈ Rd, v ∈ {1, . . . , d}. Fix τ ∈ Ω, k = 2r−1 + l,and v ∈ {1, . . . , d}. Consider the functions fv,k,τ (p) := xv(p)− xv(τ(sk)), p ∈ Rd, and

Fv,k,τ (γ) := fv,k,τ (γ(sk)) = xv(γ(sk))− xv(τ(sk)), γ ∈ Ω;

clearly Fv,k,τ ∈ Y . Using the procedure of Proposition 4.5.6(b) we get

E(Fv,k,τ , Fv,k,τ ) ≤ C1

4.7 The generator 41

Again using Lemma I.2.12 of [5] we have Kn ∈ D(E) and Kn bounded in (D(E),E1/21 )from (4.20). We apply the Banach-Saks theorem, which states that every bounded se-quence in (D(E),E1/21 ), has a subsequence whose Cesaro means converge strongly andwe get E1(Kn,Kn) −→

n→∞0 since {τk} is dense. AsKn is continuous, we may use Propo-

sition 3.5 of [5] Chapter III, from which it follows that there exists a subsequence Knk ,k ∈ N, and an E-nest Fm, m ∈ N, such that Knk converges uniformly to zero on eachFm as k → ∞. Let us follow an idea of [9], proof of Proposition 3.1. Given δ > 0 wecan find k such that Knk < δ. We have

Fm ⊂nk⋃i=1

B(τi, δ),

by (4.21), where B(x, δ) denotes the ball of radius δ centered at x. Now it follows thateach Fm is totally bounded. Fm closed and totally bounded implies Fm compact. ThusFm, m ∈ N forms an E-nest consisting of compact sets.Step 4 : For fixed τ ∈ Ω, the system of functions Fv,k,τ , v ∈ {1, . . . , d}, k ∈ N separatesthe points in Ω, Step 3 showed that there is an E-nest consisting of compact sets andthe form is a Dirichlet form by Proposition 4.5.8. Quasi-regularity now follows from itsdefinition.

Remark 4.6.2. We observe that to obtain (4.18) we need the condition (4.15). We recallthat by Proposition 4.5.6(b) we have (E, DY (E)) = (E, DZ(E)).

Corollary 4.6.3. There exists a diffusion process M associated with (E, DZ(E)).

Proof: The result is an immediate consequence of Propositions 4.5.8 and 4.6.1 usingTheorem IV.3.5 of [5].

4.7 The generator

Let ϕ(γ) have the form of (4.6) and (E, DY (E)) given by

E(F,G) =

∫〈DF,ADG〉ϕdν =

∞∑i=1

λi

∫∂SiF∂SiGϕdν, F,G ∈ Y

where DY (E) is the closure of (E, Y ) on L2(ϕν). We determine the generator A of thisform, i.e.

E(F,G) =

∫(−AF )Gϕdν, F ∈ D(A), G ∈ DY (E).

Using gn from (4.8), let Si(γ) :=∫ 1

0〈gi(s), dγs〉Rd , i ∈ N . We observe that S1(γ),S2(γ), . . .

are independent N(0, 1).

Proposition 4.7.1. We have Y ⊂ D(A). If F ∈ Y then

AF =

∞∑i=1

λi

[∂2SiF +

∂Siϕ∂SiF

ϕ+ Si∂SiF

]. (4.22)


Proof: Let F,G ∈ Y . Once again we note that the following sum is just a finite sumbecause of the particular structure of Y . We have

E(F,G) =

∞∑i=1

λi

∫∂SiF∂SiGϕdν

= lims→0

∞∑i=1

λi1

s

[∫G(γ + sSi)∂SiFϕdν −

∫G∂SiFϕdν

]

= lims→0

∞∑i=1

λi1

s

[∫G∂SiF (γ − sSi)ϕ(γ − sSi) dν(γ + sSi)−

∫G∂SiFϕdν

]

= lims→0

∞∑i=1

λi

[∫Gϕ(γ − sSi)

dν ◦ TsSidν

·(∂SiF (γ − sSi)− ∂SiF

s

)dν

+

∫Gdν ◦ TsSi

dν∂SiF ·

(ϕ(γ − sSi)− ϕ

s

)dν

+

∫G∂SiFϕ

(dν◦TsSidν − 1s

)dν

]

=

∞∑i=1

λi

∫−Gϕ∂2SiF −G∂SiF∂Siϕ+Gϕ∂SiF

d

ds

∣∣∣∣s=0

dν ◦ TsSidν

dν,

where T is the shift operator, TsSi(γ) = γ + sSi. Thus the generator satisfies

AF =

∞∑i=1

λi

[∂2SiF +

∂Siϕ∂SiF

ϕ− ∂SiF

d

ds

∣∣∣∣s=0

dν ◦ TsSidν

]

from which we can conclude F ∈ D(A). The Cameron-Martin formula gives

d

ds

∣∣∣∣s=0

dν ◦ TsSidν

(γ) = −Si(γ)

and (4.22) follows immediately.

Remark 4.7.2. Let bj denote 〈b, ej〉Rd and let gi be as in (4.8). The particular structureof ϕ gives

∂Siϕ

ϕ=

d∑j=1

1∫0

〈Dbj(γs), Si(s)〉H d(γj)s +1∫

0

〈b(γs), gi(s)〉Rd ds

− 12

1∫0

〈D‖b(γs)‖2Rd , Si〉H ds

4.8 Local first moment 43

and therefore

AF =

∞∑i=1

λi

[∂2SiF + Si∂SiF +

( d∑j=1

1∫0

〈∇bj(γs), Si(s)〉Rd d(γj)s

+

1∫0

〈b(γs), gi(s)〉Rd ds−1∫

0

〈d∑j=1

bj(γs)∇bj(γs), Si(s)

〉Rd

ds

)∂SiF

].

4.8 Local first moment

LetM =((Xt)t∈[0,1], (Pγ)γ∈Ω

)be the corresponding diffusion process defined in Corol-

lary 4.6.3, Svi denote 〈Si, ev〉Rd , and Si(γ) be as in Chapter 4.7. We recall that (A, D(A))

is the closure of (A, Y ) in the graph norm(‖ · ‖2L2(ϕν) + ‖A · ‖

2L2(ϕν)

)1/2. Thus Si ∈

Y ⊂ D(A) by definition.

Proposition 4.8.1. We have

limt→0

1

t

∫(Si(γ)− Si(τ)) Pτ (Xt ∈ dγ) = λi

[∂Siϕ(τ)

ϕ(τ)+ Si(τ)

].

Proof: According to Proposition (4.22), we have Si(γ) ∈ Y ⊂ D(A). Therefore

limt→0

1

t

∫(Si(γ)− Si(τ)) Pτ (Xt ∈ dγ) = A(Si(τ))

and by (4.22)

A(Si(τ)) =

∞∑j=1

λj

[∂2SjSi(τ) +

∂Sjϕ∂SjSi(τ)

ϕ+ Sj(τ)∂SjSi(τ)

]

= λi

[∂Siϕ(τ)

ϕ(τ)+ Si(τ)

].

For a dyadic number, sk =∑ri=1 ci · 2−i, we note that there exist N(sk) ∈ N such

that 〈(sk ∧ ·), Si〉H = 0 whenever i > N(sk).

Proposition 4.8.2. (a) Let sk ∈ [0, 1] be a dyadic number. Then xv(γ(sk)) ∈ D(A) and

Axv(τ(sk)) =

N(sk)∑i=1

λiSvi (s)

[∂Siϕ(τ)

ϕ(τ)+ Si(τ)

]= limt→0

1

t

∫(xv (γ(sk))− xv (τ(sk))) Pτ (Xt ∈ dγ).


(b) Let v ∈ {1, . . . , d}. Suppose (4.15) and∞∑p=0

λ2d2p

2p

4.8 Local first moment 45

where Γ denotes the carré du champ operator (see [1] p. 17). Thus for f ∈ Y ∩ L∞, orf ∈ Z ∩ L∞ if (4.15) holds, we have

Γ(f, f) = 2

∞∑i=1

λi(∂Sif)2.

Using Proposition 4.8.1 we obtain

limt→0

1

t

∫‖Si(γ)− Si(τ)‖2Rd Pτ (Xt ∈ dγ) = 2λi

which formally coincides with Proposition 5.1 in [4]. The same holds for linear combina-tions of Si(γ) belonging to D(E).

5Chapter 5-Notation and basic

definitions

5.1 Introduction

This chapter contains a selection of definitions and theorems that the reader is assumed tobe familiar with from earlier experience. The reader could also read this section to refreshtheir memory. The section also has a list of notation used throughout the thesis.

5.2 Definitions

Definition 5.2.1. Given a measurable space (X,Σ) and a measure µ on that space, a setA is called an atom if

µ(A) > 0

and for any measurable set B ⊂ A

µ(B) = 0.

If the measure µ has no atoms it is called non-atomic or atomless.

Definition 5.2.2. A Gaussian random variable X is centered if

E[X] = 0.

Definition 5.2.3. A topological space X is called separable if it contains a countabledense subset Y . In other words there exists a sequence {yn}∞n=1 in Y such that everynonempty open subset of X contains at least one element of this sequence.

Definition 5.2.4. A probability space (Ω,F, P ) is complete if for all B ∈ F such thatP (B) = 0 one has that for every A ⊂ B that A ∈ F.

47

48 5 Chapter 5-Notation and basic definitions

Definition 5.2.5. Let X,Y be Banach spaces. A linear operator A : D ⊂ X → Y iscalled closed if for every sequence (xn) ∈ D(A) s.t. xn → x ∈ X and Axn → y ∈ Yone has x ∈ D(A) and Ax = y.

Definition 5.2.6. Let X,Y be Banach spaces. A linear operator A : D ⊂ X → Y iscalled closable if for every sequence (xn) ∈ D(A) s.t. xn → 0 and Axn → y, one hasy = 0.

Definition 5.2.7. A set of functions {fα : α ∈ A} : X → R is called a separating setfor X or said to separate the points of X if for any two distinct elements x and y in Xthere exists a function f ∈ {fα : α ∈ A} such that f(x) 6= f(y).

Theorem 5.2.8. (Banach-Saks theorem) Let H be a Hilbert space and xn ∈ H , n ∈ Nbe bounded sequence, i.e. ‖xn‖ ≤ C for all n ∈ N. Then there exists a subsequence andan element x ∈ H such that

1

N

N∑k=1

xnk → x, as N →∞.

5.3 Notation

The same symbol can be used for different purposes. The principal notation is listedbelow, any deviations from this is explained in the text.

Symbols and Operators

〈·, ·〉H The inner product in H‖ · ‖H The norm in H· ∧ ◦ Minimum of ·, ◦A The generator1A The characteristic function of the set AB The Borel σ-algebraC∞p Smooth functions with polynomial growthD The Malliavin gradient operatorDh Directional derivativeD1,p Sobolev space (we have DF ∈ Lp)δ The divergence operatorDom The domainD(·) The domainE[·] Expected valueei Element of an ON-basisE A bilinear formEn The set of elementary functionsF A σ-algebra

5.3 Notation 49

Gα An element in the resolventH A Hilbert space. Often L2([0, 1])Hn The nth Hermite polynomial, the nth Haar functionHn The nth Wiener chaosIn Stochastic integral of order nJn The projection operator onto the nth Wiener chaosΛ The set of finite multiindicesL The generator of the Ornstein-Uhlenbeck semigroupLs Set of symmetric functionsN The natural numbers {0, 1, . . . }Ω A sample space or the space C0([0, 1);Rd)Φ Density of a normal distributionΦa A RV defined through Hermite polynomialsρ(·) The resolvent setR The real numbersΓ The carré du champ operatorSi The ith Schauder functionSi Random variable linked to SiTt An element of the semigroupµ A measure, often Lebesgue measureν A measure, often Wiener measureV [·] The varianceWt Wiener process at time tω An element of ΩY A set of cylinder functions defined on dyadic pointsZ A set of cylinder functionsZH Cylinder functions with values in H

Abbreviations and Acronyms

a.e. Almost everywherea.s. Almost surelyi.i.d. Independent and identically distributedON OrthonormalRV Random variables.t. Such thatw.l.o.g. Without loss of generalityw.r.t. With respect to

50 5 Chapter 5-Notation and basic definitions

Bibliography

[1] N. Bouleau and F. Hirsch. Dirichlet Forms and Analysis on Wiener Space. DeGruyter, 1991.

[2] B. Driver and M. Röckner. Construction of diffusions on path and loop spaces ofcompact riemannian manifolds. C. R. Acad. Sci. Paris, 315:603–608, 1992.

[3] M. Fukushima, Y. Oshima, and M. Takeda. Dirichlet Forms and Symmetric MarkovProcesses. De Gruyter, 2011.

[4] J.-U. Löbus. A class of processes on the path space over a compact riemannianmanifold with unbounded diffusion. Trans. Amer. Math. Soc., 356, 2004.

[5] Z.-M. Ma and M. Röckner. Introduction to the Theory of (Non-symmetric) DirichletForms. Springer, 1992.

[6] P.-A. Meyer. Stochastic processes from 1950 to the present. Electronic Journ@l forHistory of Probability and Statistics, 5(1), June 2009.

[7] D. Nualart. The Malliavin Calculus and related topics. Springer, 2006.

[8] D. Nualart. The Malliavin Calculus and Its Applications. AMS, 2009.

[9] M. Röckner and B. Schmuland. Tightness of general c1,p capacities on banach space.J. Funct. Anal., 108:1–12, 1992.

[10] F.-Y. Wang. Weak poincaré inequalities on path spaces. Int. Math. Res. Not., pages90–108, 2004.

[11] F.-Y. Wang and B. Wu. Quasi-regular dirichlet forms on riemannian path and loopspaces. Forum Math. Volume, 20(6):1085–1096, 2008.

51

52 Bibliography

[12] F.-Y. Wang and B. Wu. Quasi-regular dirichlet forms on free riemannian path spaces.Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12(2):251–267, 2009.

AbstractPopulärvetenskaplig sammanfattningContents1 Introduction2 Malliavin Calculus3 Dirichlet form theory4 Main results5 Notation and basicdefinitionsBibliography

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