Advances in Analysis, Probability and Mathematical Physics: Contributions of Nonstandard Analysis

Advances in Analysis, Probability and Mathematical Physics

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL

Centre for Mathematics and Computer Science. Amsterdam. The Netherlands

Volume 314

Advances in Analysis, Probability and Mathematical Physics Contributions of Nonstandard Analysis

edited by

Sergio A. Albeverio Faculty of Mathematics, Ruhr-Universitat Bochum, Bochum, Gennany

Wilhelm A. J. Luxemburg Department of Mathematics, California Institute of Technology, Pasadena, California, U.S.A.

and

Manfred P. H. Wolff Mathematical Institute, University ofTiibingen, Tiibingen, Gennany

Springer-Science+Business Media, B.Y.

Library of Congress Cataloging-in-Publication Data

Advances in analysis, probability, and mathematical physics: contributions of nonstandard analysis I edited by Sergio A. Albeverio, Wilhelm A.J. Luxemburg, and Manfred P.H. Wolff.

p. cm. -- (Mathematics and its applications; v. 314) Proceedings of an international conference held July 1992 at the

Heinrich Fabri Institut of the University of Tubingen. Inc I udes index.

1. Nonstandard mathematical analysis--Congresses. I. Albeverio, Sergio. II. Luxemburg, W. A. J., 1929- III. Wolff, M. P. H. (Manfred P. H.) IV. Series: Mathematics and its applications (Kluwer Academic Publishers) ; v. 314. CA299.82.A38 1995 515' .33--dc20 94-35635

ISBN 978-90-481-4481-5 ISBN 978-94-015-8451-7 (eBook) DOl 10.1007/978-94-015-8451-7

Printed on acid-free paper

All Rights Reserved © 1995 Springer Science+Business Media Dordrecht Originally published by K1uwer Academic Publishers in 1995. Softcover reprint of the hardcover 1 st edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

TABLE OF CONTENTS

Preface: ..................................................................... VB

Part I: Analysis

S. Albeverio, D. Guido, A. Ponosov, S. Scarlatti: Singular traces and nonstandard analysis .. ..................................................................... 3 M. Capiriski, N. Cutland: Navier-Stokes equations ............................ 20 E. 1. Gordon: Hyperfinite approximations of commutative topological groups .... 37 T. Norberg: A note on the myope topology .................................... 46 M. Oberguggenberger: Nonlinear theories of generalized functions .............. 56 H. Osswald: A nonstandard approach to the Pettis integral ..................... 75 H. Ploss: A counterexample to the spectral mapping theorem revisited from a non-standard point of view ........................................................ 91 H. Render: Nonstandard polynominals in several variables ..................... 96 T. Todorov: An existence result for a class of partial differential equations with smooth coefficients .......................................................... 107 B. Wietschorke: On the generation of topology by external equivalence relations 122 B. Zimmer: Nonstandard hulls of Lebesgue-Bochner spaces .................... 132

Part II: Probability Theory

H. Akiyama: A nonstandard approach to diffusions on manifolds and nonstandard heat "kernels ................................................................. 143 N.J. Cutland, Siu-Ah Ng: A nonstandard approach to the Malliavin Calculus . 149 M. Kalina: Ergodic transformations in AST .................................. 171 D. Landers, L. Rogge: Nonstandard characterization for a general invariance prin-ciple ........................................................................ 176 T. Lindstr~m: Andersons's Brownian motion and the infinite dimensional Omstein-Uhlenbeck process .................................................... 186 A. Ponosov: Two applications of NSA in the theory of stochastic dynamical systems ..................................................................... 200 D.A. Ross: Nonstandard methods and the space of experiments . ............... 212

Part III: Mathematical Physics

L. Arkeryd: Infinite range forces and strong V -asymptotics for the space-homoge-neous Boltzmann equation ................................................... 221 M.P.H. Wolff: A nonstandard analysis approach to the theory of quantum meanfield systems ..................................................................... 228

Subject index ............................................................... 247

V

PREFACE

During the last week of July, 1992 an international conference was held on nonstandard analysis at the "Heinrich Fabri Institut" of the University of Tiibingen in Blaubeuren, Germany. The conference was dedicated to Professor Dr. D. Laugwitz on the occasion of his sixtieth birthday. With this dedication the participants wished to honor Professor Laugwitz for the pivotal role he played in the development of the theory of infinitesimals and for his contributions to its history.

The year 1958 is an important year in the history of the theory of infinitesimals. In that year there appeared in the Mathematische Zeitschrift the paper "Eine Erweiterung der Infinitesimalrechnung", written jointly by C. Schmieden, a professor of aeronautics at the "Technische Hochschule" of Darmstadt, and D. Laugwitz. The contents of this important paper were inspired by an idea of the late Professor Schmieden to extend the real number system in a certain way to include infinitely small as well as infinitely large numbers in order to obtain a number system which would allow a representation of the Dirac a-function by means of a "point" function with values in the new number system. This was accomplished by embedding the real number system as constant sequences in the ring of sequence of reals with the natural identification for two sequences to define the same number when they are equal from a certain point onwards, a definition that in a way is akin with Cantor's definition of the reals. In this ring null sequences may be viewed as infinitesimals and those that approach infinity as infinitely large numbers. By introducing the notion of a "normal" function, Laugwitz showed in 1959 how the real functions of the calculus can be extended to normal functions on the ring. The family of normal functions, however, contains far more functions to allow the Dirac o-function to be represented by normal functions. This representation gave a satisfactory answer to the question raised by Schmieden in terms of his ideas concerning the new ring of numbers extending the reals.

It seems that at the end of the fifties the theory of infinitesimals, which had survived since the days of Leibniz and Newton only as a matter of speech, was to enjoy a survival. Around 1960, from a completely different point of view and independently of the ideas of Schmieden and Laugwitz, Abraham Robinson introduced, using the ideas of model theory, an entirely new version of the theory of infinitesimals. He showed that there exist non-archimedean totally ordered fields containing the reals that in a model theoretic sense have the same properties as the reals. The announcement of this approach to the theory of infinitesimals appeared in 1961 under the title "Nonstandard Analysis," where nonstandard referred to the nature of the new fields of numbers as defined as nonstandard models of the first order theory of the

Vll

viii Preface

reals. It turned out that the new systems of numbers were closely related. The ring of Schmieden and Laugwitz may be viewed as a reduced power of the theory of the reals and so shares with the reals fewer properties than the hyperreal number systems introduced by Robinson, being nonstandard models of the theory of the real number system.

During the last thirty years the use of nonstandard models in mathematics has taken its rightful place among the various methods employed by mathematicians. This development was in an essential way spurred on and inspired by Laugwitz. He was one of the initiators and inspiring organizers of a series of very successful "Tagungen" at the Mathematical Research Institute in Oberwolfach of which the first one was held in 1970.

Because of the controversial nature of the notion of infinitesimals throughout the history of the calculus Laugwitz, inspired by Robinson's historical account of the history of the theory of infinitesmals with which he finished his by now classical treatise on nonstandard analysis, started to examine further from a historical point of view the works of Leibniz, Newton, Euler and particularly Cauchy to trace the rise and fall of the use of the infinitesimal arguments in the calculus. From these historical investigations a much clearer picture has emerged of the role infinitesimals has played in the development of the calculus l .

The contributions in this book, dedicated to Professor D. Laugwitz, have been selected by the Editors to present a panoramic view of the various directions in which nonstandard analysis is advancing and we hope may be a source of inspiration for the future.

The first group of contributions consists of papers in the areas "Analysis, Topology and Topological groups". In the contribution by S. Albeverio, D. Guido, A. Ponosov, S. Scarlatti, on "Singular traces and nonstandard analysis", singular traces on the space of compact operators in a separable Hilbert space (of importance e.g. in non commutative geometry) are characterized in terms of nonstandard analysis, extending results of Dixmier and Varga. Corresponding ergodic states are also discussed, relating to previous work on Banach-Mazur limits by methods of nonstandard analysis. In their article ''Navier-Stokes equations" M. Gapinski and N. Gutland prove the existence of weak solutions of a very general equation of this type. Moreover they succeed also in solving the correspondent stochastic differential equation. The contribution by E.!. Gordon, "Hyperjinite approximations of commutative topological groups", presents a nonstandard approach by hyperfinite approximations to harmonic analysis on arbitrary locall compact abelian groups having a compact open subgroup. T. Norberg in "A note on the myope topology" discusses the identy of two natural topologies, the myope and Lawson's one, on the collection of compact subsets of a locally compact Hausdorff space. A characterization of the monads of the myope

lSee the companion volume, also dedicated to Professor D. Laugwitz, "Analysis and Geometry", edited by B. Fuchststeiner and W. A. J. Luxemburg, BI Wissenschaftsverlag, Mannheim 1992.

Preface ix

topology is given, opening the way to the study of capacities in this setting. M. Oberguggenberger in "Nonlinear theories of generalized functions" presents an overview of Rosinger-Colombeau's theory, pointing out its relations with nonstandard analysis. One of the main advantages of this theory is its ability to handle some of the multiplication problems for distributions. Applications to non linear partial differential equations are also discussed. The contribution by H. Osswald, "A nonstandard approach to the Pettis integral", presents lifting theorems for stochastic processes with values in a nonseparable locally convex space. Various applications are given including the existence of solutions for nondeterministic Peano-Caratheodory differential equations. H. Ploss in "A counterexpample to the spectral mapping Theorem revisited from a nonstandard point of view" uses non standard analysis methods to provide a natural explanation for the nonvalidity of the spectral mapping theorem for the exponential function. H. Render in "Non standard polynomials in several variables" extends results of C. Impens on nonstandard polynomials to the case of several variables and relates polynomial approximations in function theory with complex and real micro continuity of non standard polynomials. In his contribution "An existence result for a class of partial differential equations with smooth coefficients" T. Todorov proves the existence of a solution for a class of linear partial differential equations with COO-coefficients in a space of generalized functions larger than the space of Schwartz distributions. B. Wietschorke in "On the generation of topology by external equivalence relations" extends Reeken's theory of external equivalence relations in Nelson's Internal Set Theory to vector spaces. He gives a necessary and sufficient condition for the smallest monad containing the external set of infinitesimals to be the monad of a topology. Applications to locally convex spaces are given. B. Zimmer in "Nonstandard hull of Lebesgue-Bochner spaces" discusses the structure of the nonstandard hull of Lp(ll, E}, showing that between it and Lp(P., E} (with E resp. p. the nonstandard hull of E resp. the Loeb measure associated with /-L) there is a space isometric to the space Mp(P., E} of E-valued functions that have an SY -integrable-lifting. She also shows in particular that the Bochner integral extends naturally to Mp(P., E}. The second group consists of papers in Probability Theory. The contribution by H. Akiyama "A nonstandard approach to diffusions on manifolds and nonstandard heat kernels" uses internal methods to give nonstandard constructions for diffusions and heat kernels on manifolds. N.J. Cutland and Siu-Ah Ng in "A nonstandard approach to the Malliavin Calculus" give an intuitive approach to the Malliavin calculus for the classical Wiener space, showing in particular that the basic operators of this calculus have natural descriptions as classical differential operators on a nonstandard space R for some infinite natural number N. This has applications to stochastic differential equations. Ergodicity of set transformations within the Alternative Set Theory is studied in the contribution by M. Kalina "Ergodic transformations in AST". In fact ergodic transformations are shown to be homomorphic to a cycle of some suitable length. In their contribution "Nonstandard characterization for a general invariance prin-

x Preface

ciple" D. Landers and L. Rogge show that every invariance principle of probability theory is equivalent to a nonstandard construction of internal S-continuous processes, which all represent, modulo an infinitesimal error, the limit process. The contribution by T. Lindstr¢m "Anderson's brownian motion and the infinite dimensional Ornstein-Uhlenbeck process" presents a natural extension of Anderson's construction of Brownian motion as infinitesimal random walk to the infinite dimensional Ornstein- Uhlenbeck process. A Brownian sheet representation is also given, as well as two other hyperfinite constructions, which are discussed in their respective merits. In "Two applications of NSA in the theory of Stochastic dynamical systems" A. Ponosov presents a new infinite dimensional multiplicative ergodic theorem for generalized cocycles. The proof shows how methods of nonstandard analysis permits to extend Oseledec's methods to the infinite dimensional case in a natural way. Another application of nonstandard analysis (Loeb measures) to a proof of a new result about standard objects is also given, namely a construction of stationary solutions for nonlinear infinite dimensional stochastic differential equations with a monotone principal part, extending results of E. Browder. The contribution by D.A. Ross "Nonstandard methods and the space of experiments" gives nonstandard proofs of two results of the theory of statistical experiments. The first one shows that the space of experiments is compact with respect to a certain weak topology. The second proof gives the validity of a criterium for the equivalence of experiments.

The third group of papers concerns applications of nonstandard analysis to Mathematical Physics. L. Arkeryd in "Infinite range forces and strong Ll-asymptotics for the space-homogeneous Boltzmann equation" illustrates the strength of the applications of nonstandard analysis in the kinetic gas theory by discussing the strong L1-convergence in time to Maxwellian limits for standard solutions of the Boltzmann equation starting far from equilibrium. In particular the convergence is proven for the space homogeneous Boltzmann equation without angular cut off, in a situation for which no nonstandard proof is known as yet. In "A nonstandard analysis approach to the theory of quantum meanfield systems" M.P.H. Wolff develops an approach to the theory of (quantum mechanical) mean field systems in which the relevant algebras of observables are quotients of subalgebras of a hyperfinite tensor product of the relevant one-particle algebra. In particular, this underlines the statistical character of meanfield phenomena. A dec tailed discussion of meanfield dynamics and of the relation between the quantum and classical case is also given. The present approach also serves as a basis for a theory of fluctuations, to be discussed in subsequent work.

We would like to thank Professor Michael Hazewinkel and the Publishing House Kluwer for their offer to publish this book in their Mathematical Sciences Series.

The Editors

August 10, 1994

Part I:

ANALYSIS

SINGULAR TRACES AND NONSTANDARD ANALYSIS

S. Albeverio Fakultat fiir Mathematik, Ruhr-U niversitat Bochum

D-44780 Bochum, SFB-237 Essen-Bochum-Diisseldorf, CERFIM, Locarno

D. Guido Dip. di Matematica, Universita di Roma Tor Vergata, Italy

A. Ponosov t Fakultat fiir Mathematik, Ruhr-Universitat Bochum

D-44780 Bochum

S.Scarlatti Dip. di Matematica, Universita di L' Aquila, Italy

Abstract

We discuss non trivial singular traces on the compact operators, extending some results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states using tools of non standard analysis.

1 Introduction

Let H be a complex separable Hilbert space and let B(H) be the Banach algebra of all bounded linear operators on H. It is well-known that every non trivial normal (Le. continuous in some sensej see Section 2 for precise definitions) trace on B(H) is proportional to the usual one (see e.g. [6]). On the other hand Dixmier proved in 1966 [7] that there exist non trivial traces on B(H) which are not normal (moreover they are "singular" in the sense of definition 2.2 below).

The basic idea of Dixmier for constructing singular traces was to consider the compact operators for which the usual trace diverges at a given suitable rate and to associate to any such compact operator a suitable element of the Banach space

tSupported by the Deutsche Forschungsgemeinschaft

3

4 S. Albeverio et al.

fOO{IN) of all bounded sequences. The singular trace is then obtained evaluating on such elements a state on fOO{IN) which is invariant under "2-dilations".

The importance of singular traces is well-known now due to their applications in non commutative geometry and quantum field theory (see e.g. [3]).

The general question which operator ideals in B(H) support traces has been recently studied by Varga [16] who used different kinds of states on fOO(IN).

In our two preceding papers [2], [3] we gave explicit formulas for Dixmier-type traces and introduced a new class of singular traces.

The present paper is organized as follows. In the first part of Section 2 we deal with the problem of describing which kind of sequences can be "summed" by a non trivial singular trace. We shall give a complete answer to this problem, generalizing some results by Varga [16].

In the second part of Section 2 we generalize the construction of non-normal traces given by Dixmier in [7].

In Section 3 we describe ergodic states giving rise to both kinds of singular traces introduced in Section 2. The basic technique we use is related to non standard analysis (NSA). The Section 3 also involves the representation of Banach-Mazur limits by NSA. Such representation have been discussed before - e.g. in [12], [13].

In Section 4 we work out explicitly the computation of the Dixmier traces of an operator again using the NSA framework in an essential way.

2 Singular traces and generalized eccentric operators

The content of this section is purely standard. So we will omit all the proofs which can be found in [4]. We start recalling some basic definitions and results.

Let.n a von Neumann algebra and n+ the cone of its positive elements. A weight on n is a linear map

cf> : n+ -+ [f, +00]

Any weight can be extended by linearity on the natural domain given by the linear span of {T E n+Icf>(T) < +oo}

A weight T which has the property:

T(T*T) = T(TT*) \lTEn

is called a trace on n. The natural domain of a trace is a two-sided ideal denoted by IT' For instance the natural domains of the trivial traces on n given by T == 0 and T == +00 are

Singular traces and nonstandard analysis 5

respectively the ideals Rand {O} while the usual trace on B(H), the bounded linear operators on a complex, separable Hilbert space H, is associated with the ideal L1(H) of the trace class operators.

A weight 4> on R is called normal if for every monotonically increasing generalized sequence {Teo a E I} of elements of R+ such that T = sup", T", one has

4>(T) = lim4>(T",) '"

From now on the von Neumann algebra R will be fixed to be B(H).

A classical result [6J concerning normal traces on B(H) is the following:

Theorem 2.1 Dixmier Every non trivial normal trace on B(H) is proportional to the usual trace.

By a theorem of Calkin (see [10]), each proper two-sided ideal in B(H) contains the finite rank operators and is contained in the ideal K(H) of the compact linear operators on H. Therefore all traces on B(H) live on the compact operators, and the following definition makes sense:

Definition 2.2 A trace 7 on B(H) will be call singular if it vanishes on the set F(H) of finite rank operators.

Proposition 2.3 Any trace 7 on K(H) can be uniquely decomposed as 7

71 + 72, where 71 is a normal trace and 72 is a singular trace.

In view of this proposition, in the rest of the paper we shall restrict our attention to the singular traces.

For T a compact operator on H, {!-Ln(T) }::"=1 will denote the non increasing sequence of the eigenvalues of ITI with multiplicity.

n

We shall also set (J'n(T) == 2: !-Lk(T). k=1

Definition 2.4 Let T be a compact operator. We call integral sequence ofT the sequence {Sn(T)}::"=o which is an indefinite integral (w.r.t. the counting measure) of {!-Ln(T)}::"=l, i.e. Sn(T) - Sn-l(T) = !-Ln(T), n ~ 1, and such that

_ {O T ~ Ll(H) So(T) = -tr(T) T E Ll(H)


Notice that if T ~ Ll(H), Sn(T) = un(T), n ~ 1, while if T E L1(H), then Sn(T) = un(T) - tr(T} --+ O. We also give the following

Remark 2.5 If T does not belong to L1 (H) and r is a trace which is finite and non-zero on ITI then r is necessarily singular, that is, the existence of traces which are non trivial on T is equivalent to the existence of non trivial singular traces on T. Since for T E L1(H} the existence of a non trivial trace is obvious, it follows that the relevant question is not the mere "traceability" of a compact operator T, but the existence of a singular trace which is non trivial on ITI.

Let us also notice that a trace r is finite on ITI if and only if the principal ideal I(T), i.e. the (two-sided) ideal generated by T in B(H}, is contained in IT'

Lemma 2.6 Let T be a compact operator. The following are equivalent

[1} 1 is a limit point of the sequence { ~:l;:l}:o

[2} There exists an increasing sequence of natural numbers {pd such that 1· Skp.(T) 1 1mk-+oo SPk (T) =

Definition 2.7 A compact operator T which satisfies one of the equivalent pro-perties of Lemma 2.6 will be called generalized eccentric.

Remark 2.8 The class of generalized eccentric operators which are not in L1(H} coincides with the class of eccentric operator considered in [16}.

We can now state the main result of this section.

Theorem 2.9 Let T be a compact operator. Then the following are equivalent:

[1} There exists a singular trace r such that 0 < r(ITJ) < +00.

[2} T is generalized eccentric.

As we have already mentioned the proof of the theorem can be found in [4]. Nevertheless we are going to discuss here some possible procedures to construct singular traces on K(H} because it is important for the non standard part of this work. Basically, all singular traces in question can be represented as follows:


AEI(r) (1)

where 1/; is a generic non normal state on foo (IN) == fOO.

Recall that the state is just a linear bounded positive functional with the property 1/;( {1}} = 111/;11 = 1. It is easy to see thatfor such a 1/; the function To/> has all properties of traces with the only exception of additivity which requires some more conditions to be imposed upon. In the literature there are two ways to do this. Although both of them were introduced in more particular situations than we are dealing with they can be adapted to our situation too. The first way is due to Dixmier [D] and consists in choosing 1/; to be 2-dilation invariant (therefore 1/; vanishes on the set Co of the infinitesimal sequences, hence it is not normal):

Va E fOO.

In this case the generalized eccentric operator T must satisfy the stronger condition:

(compare with the conditions given in Lemma 2.6}.The second way of getting addi

tivity was suggested by Varga [16] who considered non normal states supported by "fast" sequences, i.e. the states of the form:

(2)

where cp is an arbitrary non normal state on fOO and the sequence nk is given by nk == kPk with Pk defined in Lemma 2.6. We now formulate briefly two results on

the existence of singular traces.

Theorem 2.10 Let T be a generalized eccentric operator. The functional To/>

defined in (1) and (2) linearly extends to a singular trace on the ideal I(r).

Theorem 2.11 If1/; is a two-dilation invariant state andlimnEIN S;:(~l = 1, then To/> is a trace on I(r). Moreover, in this case formula (1) gives rise to a singular trace (which will be denoted by To/> as well) even on the (larger) ideal

Im(T} == {A E K(H}I { ~:~~~} E foo }

We would like to point out that, when T rt. Ll(H}, the ideal I.n(T} is a maximal norm ideal in the sense of Shatten [15] (see also [10]).

8 S. Albeverio et aL

Remark 2.12 We should stress that we did get new types of singular traces. Of course, if the operator T ~ L1(H) then the traces described in Theorems 2.10 and 2.11 are exactly the traces discussed by Dixmier in [7} and by Varga in [16} respectively, and they vanish on L1 (H).. In the case TELl (H) our Theorems 2.10 and 2.11 produce new classes of singular traces, which in a sense are the inverse images inside L1(H) of the class of Varga-type traces and of the class of Dixmiertype traces respectively. We notice also that if the trace 7'2 in the decomposition of proposition 2.3 does not vanish, it should be of this new type, i.e. it should not vanish on £1 (H).

The existence of new Dixmier-type traces was announced in [AGPS 2].

3 Two-dilation invariant states and ergodicity

The main problem we are going to discuss in this section concerns extremal (ergodic) states which give rise to singular traces. In our opinion it is non standard analysis which supplies us the most convenient tools for this purpose. We first briefly discuss extremal states corresponding to the Varga-type traces described in Theorem 2.11. Let us denote by ~{nk} the set of all extremal points in the set of non normal states

of the form (2).

Proposition 3.1 The set ~{nd consists of the states

for some mE ·INoo = ·IN - IN.

Proof: It immediately follows from the fact that extremal states on foo are all of the form

(3)

Since additionally cp must vanish on the set Co then m becomes infinitely large .•

Remark 3.2 Of course, instead of infinitely large numbers one can equivalently use the Stone-Cech compactification IN of IN and the isomorphism foo(IN) ~ C(IN) given by the Gelfand transform in order to describe extremal states in Proposition 3.1. Namely, they will be given by Dirac measures supported by the set IN \ IN. On the contrary, the classification of ergodic 2-dilation invariant states does require NSA (see e.g. the next remark).


Now we come to the much more difficult problem of classification of two dilation invariant states. First, we remark that in order to prove the existence of such states, Dixmier invoked the amenability of the affine group. As promised, here we shall adopt an alternative point of view which relies on the use of NSA and related methods (see e.g. [11], [1]).

Theorem 3.3 The map w -+ CPw, w E "INoo, defined by

CPw ( a) == 0 (~ t "a2k ) k=l

(4)

takes values in the convex set of 2-dilation invariant states over £00.

Proof: Let bn == ~ ~;=1 a2n • Since {an} is bounded {bn} is also bounded so that cpw(a) = O("bw) is well defined for all w. Obviously, CPw is a state. It is also 2-dilation invariant since:

A consequence of this theorem is that an explicit formula for the previously introduced traces can easily be given.

Corollary 3.4 If T is an operator verifying limn ~ = 1 and w is an infinite hypernatural then

(A) = ° (..!:. ~ "S2 k (A)) 'T", - w~OSk(T)

k=l 2

(5)

is one of the singular traces described in Theorem 2.12.

The proof of this corollary follows immediately from Theorems 2.12 and 3.3.

There is a simple generalization of the formula (4) which describes 2-dilation invariant states. If j E "IN and n E °INoo the map

(6)

is a 2-dilation invariant state over £00 and therefore gives rise to a singular trace. Since any hypernatural j can be written in a unique way as a product of an odd

number and a power of 2,j = (2m - 1)2k -1, we may rewrite the previous states as


<'ok,m,n(a) = 0 (~ E *a(2m-l)2i-1) .=k+l

k, mE *IN, n E *lNoo (7)

In the rest of the section we shall study states of the form (7) in relation to the problem of ergodicity.

Let ~ : IN -+ IN the multiplication by 2, ~* the corresponding morphism on lOO(lN), ~*( {an}) = {a2n}, we shall say that the state <,0 is ~-invariant if <,0 0 ~* = <,0.

We shall give necessary conditions for extremality in the (convex compact) set of ~-invariant states in terms of NSA.

It is known (see e.g. [8], p.1l3) that the states on lOO(IN) can be identified with the finitely additive probability measures on IN, therefore we shall denote any such a state by Il, and the notation Il(A) with A c IN makes sense.

Moreover extremality of a ~-invariant state Il can be expressed in terms of ergodicity of Il seen as a measure, i.e. for each A c IN such that ~A = A, Il(A) = 0 or 1.

Remark 3.5 Using the Stone-tech compactification IN of IN and the isomorphism lOO(lN) ~ C(lN) given by the Gelfand transform once again we get an identification of the states on lOO(lN) with the u-additive probability Radon measures on IN. On the other hand a transformation on IN extends to a continuous transformation on IN. We shall denote with 71, ~ the measure and the transformation on IN induced by Il and ~ respectively. It turns out that ergodicity of71 is equivalent to ergodicity of the finitely additive measure Il. This equivalence can be shown using well known criteria for ergodicity (see e.g. [10]).

Remark 3.6 Let us consider the correspondence TJ : IN x IN -+ IN defined by (m, n) -+ (2m - 1 )2n~1, which is a bijection. It induces an isomorphism: TJ* : lOO(IN) -+ lOO(lN x IN) given by

(TJ*a)m,n == a'1(m,n),

~. == TJ;l~TJ* becoming the translation T in the second variable: ~*(m, n) = (m, n+ 1) == (m, Tn). It might be thought that the isomorphism TJ* gives rise to the splitting of the dynamical system (IN, ~) in a product of two dynamical systems (IN, id) and (IN, if), thus furnishing a standard approach to the considerations we shall make below. Unfortunately this is not true since the spaces IN x IN and IN x IN are not homeomorphic (see e.g. [9]). Our idea is to exploit the features of non-standard analysis, in particular the nice functorial property *IN x* IN =* (IN x IN).

Let MA denote the set of extremal ~-invariant (i.e. ergodic) states on lOO(lN).

Proposition 3.7 Any Il E MA coincides with one of the states <,Ok,m,n for some k, m E* IN, n E* lNoo •


Proof: Evidently, for each state /J- and each finite dimensional E E fOO(IN) == foo

there exist numbers s(E) and Aj(E) such that

[1] /J-(a) = ~;~) Aj(E)aj (Va E E),

[2] Aj(E) ~ 0,

[3] ~;~) Aj(E) = 1.

For every hyperfinite dimensional space E (see e.g. [1], p.55), foo c E c *foo , there is therefore an internal set PIt ... ,A.} satisfying (ii) and (iii) and such that

• /J-(a) = */J-(*a) = L A/aj

j=1

If /J- is 2-dilation invariant, then for all finite n

(Va E fOO)

As it can be easily checked, for each finite dimensional space F C f oo , and each n E IN we have

1 n } + 1*<Pk,m,n(a) -;;; ttaj2.-11 ~ 0 (8)

where j = (2m - 1)2k-l. By saturation, there are a hyperfinite dimensional space F , foo C F C *fOO and a number n E* INoo such that (8) remains valid. This means that

*/J-(a) ~ L A/'Pm,k,n(a) (j = (2m - 1)2k- 1) (9) j=1

for all a E foo and immediately implies that the set of all T-invariant states coincides with the closed convex hull coM of the set M == {/J-k,n I k E* IN, n E* 1N00 }.

Finally, we show that M is closed which, according to [8], p.708, would imply the

inclusion Mil C M. Consider a directed set r c* 1N3 and assume that 'Pk,m,n~/J-. Clearly, for every finite dimensional E C foo there exists a subsequence {(ki' mi, ni)} E r such that


"CPk;,m;,n;(a) ~ "J.L(a) forsome i E"lNoo and for all a E "E, lIa!!:::; 1.

If J.L is 1::.. invariant then by saturation the latter relation holds for a certain hyperfinite dimensional E J fOO with k E" IN and n E" 1N00 • This concludes the proof. •

Corollary 3.8 The problems of description of extremal 2-dilation invariant and extremal translation invariant states are equivalent.

Proof: Recalling the map 'TJ" given by ('TJ"a)m,n = a'l(m,n) where 'TJ"(m,n) = (2m-1 )2n - 1 and applying the proposition just proved we conclude that for each fixed m the states CPk,m,n O'TJ" coincide with the translation invariant states J.Lk,n on foo defined by

( 1 k+n ) J.Lk,n(a) =0 ~ L "ai ,

i=k+l

so that any J.L E 'TJ;l(Mf).) is contained in one of the sets {om®v!v E MTnM} where Om is given by om(a) = 0 ("am), mE" IN, MT stands for the set of extremal translation invariant states on foo and M = {J.Lk,n! k E" IN, n E" IN oo}. On the other hand it is known (see [12J, [13J, or also [2J where the result was proved independently) that MT C M which completes the proof. •

Remark 3.9 Of course, the representation given in Proposition 3.7 is not unique due to Corollary 3.8 and the following trivial

Proposition 3.10 If k;:l ~ 0 and 7 ~ 0, then J.Lk,n = J.L1,p'

Now we formulate the main result in this section more precisely.

Theorem 3.11 If J.L is an extremal 2-dilation invariant state on fOO then J.L = CPk,m,n (with CPk,m,n defined by (7)) for some mE" IN and infinitely large hypernaturals k and n such that ~ ~ o.

Proof: By virtue of Corollary 3.8 it suffices to show that if v E MT then v = J.Lk,n for some k, n E" 1N00 • We first prove that k E" 1N00 • Suppose it is not the case and k is finite. Without loss of generality we can assume k = 1, and, due to proposition 3.10, n = 2m. If we show that J.Ll,n # J.Ll,m then the representation J.Ll,n = ~(J.Ll,m + J.Lm,m) implies J.Ll,n is not extremal.

For b = m2-P , choose p E" 1N00 such that 0 < °b < 00 and consider a non decreasing sequence {bj } such that "b2p ~ b. Define a sequence {cJ of natural numbers by putting Cj = [2 j bj J ([.J denotes the integer part). Since


one gets Cj+1 ~ 2cj - 2. At the same time,

= ° (*C2P - m) ° (22P ) = 0 ([22P*b2P] _ b) ~ 22p m 22p °b

< o~ O(*b2p - b) = O.

By the same reason, 'c2P: 1 -n ~ O. Applying proposition 3.10 once again we obtain

1 n 1 'C2p+l

- L*ai ~ -- L 'ai n i=l C2p+1 i=l

(Va E f"") (10)

Now we introduce a set B = U~1[C2q-b C2qj C IN and a sequence X = {Xi} where Xi = 1 for i E Band 0 otherwise. By (10),

f.L1,n(B)

It remains to prove that f.L1,m(B) :I O. In order to see this, let us observe that *Xi = 1 for i E ['C2p-b 'C2p] and that *C2p - 'C2p-1 ~'C2p-1 - 2. Hence, Hi I 'Xi = I} ~ ~'C2p - 1 and

so that f.L1,n is not extremal. We continue the proof assuming k E* INoo and I 'if!, 0 or, equivalently, ° (~) < 00. We have to show that again f.Lk,n is not extremal. First we notice that


and analogously,

-k + [Jk]2 ~ 0, n

k + n - [Jk+11]2 ~ 0, n

k+n-[v'k+n+1]2 ----'--------'- ~ O.

n

Applying now proposition 3.10, one can assume that k = (2r)2, k + n = (2s)2.

Let us introduce a set C == U~1[(2i - 1)2, (2i)2] and observe that

(C6.TC) 0 (Hi I i2 E [k,k + n]}) = 0 (Hi 12r:::; i:::; 2S}) = J.Lk,n :::; n (2s)2-(2r)2 O.

Extremality of J.L would imply, therefore, that J.Lk,n( C) should have been equal to o or 1. On the other hand,

o (~;i;1 ((2i)2 - (2i - 1)2))

= 0 (( 4s + 1 + 4r + 3)( r - s)) = ~. 8r2 - 8s2 2

This contradiction implies the result .•

Remark 3.12 The necessary conditions in theorem 3.11 are surely not sufficient. To see this, one can consider a state J.L == J.L4P-n,n for arbitrary p, n E* lNoo • If n ~ 22p- 1 , then non extremality of J.L follows from the theorem. Otherwise, we may introduce a set A == U~l (22q-1, 22q] which is easily seen to be J.L - a.e. T-invariant, but J.L(A) = ~.

Now we give a Corollary which relates the results of this section with the description of the Dixmier-type traces (for the proof cf. [2]).

Corollary 3.13 Let T be a Dixmier-type trace on the ideal I".(T) (see section 2). Then T is in the closure of the convex hull of the family

{Tk,m,n I m E·IN, k,n E·lNoo , (~) ~ O},

where Tk,m,n is the trace associated with the state TCPk,m,n via formula (1) on the same domain Im(T).

Remark 3.14 The states J.Lk,n can be seen intuitively as averages on intervals of the set ·IN. This suggests to call ergodic all the intervals associated with ergodic states.


Then it is easy to show that if the interval I is ergodic and a subinterval J is such that m ¢ 0 (where III denotes the length of I), then J.Lr = J.LJ. A sketch of the proof is the following: let I = 10 u J U II be a partition of I into subintervals. It turns out that

0(1101) 0 (IJI) 0 (Ihl) lIT J.Lro + TIT J.LJ + lIT J.Lh = J.Lr,

hence, by the ergodicity of I, J.Lr = J.LJ.

4 A computational example

We shall now discuss some advantages of representing singular traces by means of NSA.

A remarkable advantage lies, in our opinion, in the increased computability of the value of a singular trace on a given operator when such a trace is parameterized by some infinite number.

In what follows, we shall work out an example in which we explicitly calculate the value of the Dixmier trace of an operator, even though it depends on the nonstandard parameter.

To this aim we shall make use offormula (5) choosing a compact operator T such that S .. (T) = log n. The choice of "summing" logarithmic divergences has extensively been used by Connes in some applications to non-commutative geometry [3].

Let q ~ 1 be a fixed natural number, we consider a positive compact operator Aq whose sequence of eigenvalues (A .. I n = 3,4 .... ) is defined in the following way: let (nk I k = 0, 1, ... ) be an unbounded increasing sequence of natural numbers (with no == 1) whose explicit dependence on q will be given below. For n E (2"·,2"·+'], we define

nk+1 - nk

2"0+' - 2". 2m

(11)

For m ~ 2 we consider the sum U2m := ~ Aj. Let nlc < m::; nlc+1! then we have j=3

(12)

since

10-1 2Rr+1

U2m = E E Aj +


Now let p > 1 and hence ns < p $ n s+1 for some s, we have

~ L: ~ = _1_ . ~ L: L: 0"2m + L: 0"2m P (S-l nk+l P)

102m 102m m p m=l g g p k=O m=nk+1 m=n.+1

(13)

We now proceed to estimate the sums appearing on the r.h s. of (13). By means of

(12) we have

(14)

We notice that the following equalities hold

(15)

(16)

'f: 0"2m = [nk log nk+1 + 0 (1- ~)] [1 + 0 (: •. )] (17)

m=nk+1 m nk nk+1 ..

under the assumption 0 (':k) ~ 0 (2il:~1) .

To verify such a condition we fix the initial sequence (nklk = 0,1, ... ) to be of the form nk := 2kq , where q E IN.

Formula (17) takes then the form:

nk+l

L: O"~m = [2 kqq log 2 + 0(1)] [1 + O(Tkq)]

m=nk+1

(18)

Therefore we obtain

1 s-l nk+l 0"2m 1 ( 2sq - 1 ) -- - - -- 10 2-- + 0 S p log 2 £; m~+l m - q log 2 P g 2q - 1 ( )

(19)


Now, by definition, taking p E ·lNoo , we have

(20)

o c;q) 2q ~ 1 + 0 (p l~g 2 mtH · c:) ) where, in the last equality, we have used (19) and the fact that 0 (:) = o. To end the computation of the Dixmier trace of Aq we need to evaluate the second term in the r.h.s. of (20).

We have, for p E IN,

(21)

= _1 (ns) (1 + 0 (~)) ·(log~ + 0 (~)) + log 2 p ns n. n.

+ P(IO~2)2 C:::~ = ;:.) (: - ~:) (1 + 0 (~.)) Hence, by estimates similar to the previous ones, and taking p E ·lNoo we obtain

o (PI~g2 mtH · CT:) ) = IO~2 0 c;q) log 0 (;'q) (22)

From (20) and (22) it follows

(23)

where t := 0 (2;q) . In general t can take any value in the interval [2-q, 1]. In particular, in the case p = 2sq+r, 1 ~ r ~ q, formula (23) becomes

(24)

18 S. Albeverio et aL

References

[1] Albeverio, S., Fenstad, J.E., H!2Iegh-Krohn, R, Lindstr!2lm, T.: Non standard methods in stochastic analysis and mathematical physics., Acad. Press, Orlando (1986).

[2] Albeverio S., Guido D., Ponosov A., Scarlatti S., "Dixmier traces and non standard analysis", Proceedings of the III International Conference "Stochastic Processes: Physics and Geometry", Locarno June 1991, Edts. S. Albeverio et al., World Scientific Pub!. (1993).

[3] Albeverio S., Guido D., Ponosov A., Scarlatti S., "Non standard representation of non normal tracd', Proceedings of the German-French ZiFMeeting "Dynamics of the Complex and Irregular Systems" Bielefeld, Germany, December 1991, Edts. Ph. Blanchard et al. World Scientific Pub!. (1993).

[4] Albeverio S., Guido D., Ponosov A., Scarlatti S. "Singular traces and compact operators.I.", Preprint, Ruhr-Universitat Bochum, April 1993.

[5] Connes, A.: Geometrie non commutative, Intereditions, Paris (1990).

[6] Dixmier, J.: Von Neumann algebras, North Holland (1981)

[7] Dixmier, J.: Existence de traces non normales, C.R Acad. Sci. Paris 262, (1966).

[8] Edwards, RE.: Functional analysis, theory and applications, New York (1965).

[9] Glicksberg, I.: Stone-tech compactifications of products, 'Trans. AMS 90 (1959).

[10] Gohberg, I., Krein, M.G.: Introduction to the theory of non-selfadjoint operators, Mos- cow (1985).

[11] Hurd, A.E, Loeb, P.A.: An introduction to non-standard real analysis, Acad. Press, Orlando (1985).

[12] Keller, G., Moore, L.C., Jr.: Invariant means on the group of integers, in: Analysis and Geometry, Ed. B. Fuchssteiner and W.A.J. Luxemburg, Wissenschaftsverlag, (1992) 1.

[13] Luxemburg, W.A.J.: Nonstandard hulls, generalized limits and almost convergence, in: Analysis and Geometry, Ed. B. Fuchssteiner and W.A.J. Luxemburg, Wissenschaftsverlag, (1992) 19.

[14] Mane, R: Ergodic theory and differentiable dynamics, Springer, BerlinHeidelberg-N.Y. (1987).


[15) Shatten, R.: Norm ideals of completely continuous operators, Springer, Berlin - Heidelberg - N.Y. (1970).

[16) Varga, J.V.: Traces on irregular ideals, Proc. of Am. Math. Soc. 107, (1989) 715.

NAVIER-STOKES EQUATIONS

M. CapiIiski Instytut Matematiyki Vniwersytet Jagiellonski

VI. Reymonta 1, 30-059 Krakow, Poland

N.J.Cutland Dep. Pure Mathematics, Vniv. of Hull,

Cottingham Road, Hull HU 6 7RX, England

Abstract

The methods of nonstandard analysis are applied to the study of NavierStokes equations. We give a construction of weak solutions, solve general stochastic Navier-Stokes equations, and show how to obtain statistical solutions in the general stochastic case.

1 Introd uction

The N avier-Stokes equations

{

Ut - v~u + (u, V)u = f + Vp divu = 0 Ul8D = 0

(1)

describe the time evolution of the velocity field u = (Ul> ••• , un) of an incompressible viscous fluid. If n = 3, which is most interesting from the point of view of physical interpretation, this is a system of four equations. The unknown functions are Ul> U2,

U3, where Ui : D X [0, (0) -+ IR (D C 1R3 is a bounded domain with C2 boundary) that represent the components of the velocity vector, and p : D x [0,(0) -+ IR representing the pressure. The equation div U = 0 is the incompressibility condition. It allows us to eliminate the pressure and to reduce the problem to the study of the time evolution of u. We equip this system with the Dirichlet boundary condition.

Nonstandard analysis has proved to be a powerful tool in tackling a wide range of problems concerned with Navier-Stokes equations. The first step was the proof of existence of weak solutions given in [5]. This proof is substantially simpler than the

20

Navier-Stokes equations 21

standard ones and pointed the way to further results. The most important ([4]) was the existence of solutions to general stochastic Navier-Stokes equation of the form

du = (vLlu - {u, V)u + f(t, u))dt + g(t, u)dWt .

This solved a problem which was open for 18 years: in 1973 Bensoussan and Temam wrote the first paper [2] on stochastic Navier-Stokes equation with constant g. This equation has drawn the attention of many authors (see [13], [14], [9], for example) but the general case offeedback in 9 was first treated by Bensoussan [1] for dimension 2, simultaneously with the above mentioned paper covering the 3-dimensional case. Another nonstandard existence result [6] allows some special unbounded operators to appear in the noise term.

The nonstandard construction of solutions is based on a certain hyperfinite-dimensional representation DIN of the function spaces involved. Using the transfer principle we solve a system of ODE's for the deterministic case (SODE's in the stochastic case). The solution is then shown to be nearstandard in a suitable topology. The final step is to show that the standard part satisfies the appropriate N avier-Stokes equation.

We give a complete proof for the deterministic case in Sect.3 and outline the main steps of proof in the stochastic case in SectA. In both cases the theorems given are stronger than those in [5], [4]. We conclude with a proof of existence of socalled statistical solutions for the stochastic equation (hence in particular we have existence of statistical solutions for the deterministic one).

2 Preliminaries

For a fixed domain D C Rn, n -:; 4, with boundary of class C2 let DI be the closure of the set

v = {u E CO'(D,Rn ): divu = O}

in the £2 norm lui = (u,u)1/2, where

(u,v) = t 1 ui(x)vi(x)dx. i=l D

The letters u, v, w will be used for elements of DI.

The symbol V denotes the closure of V in the norm lui + lIuli where Ilull = ((u, u))1/2 and

~(au av) ((u, v)) = L.J ~'~ . i=l x, x,

DI and V are Hilbert spaces with scalar products (.,.) and ((', .)) respectively, and I . I -:; ell . II for some constant c.

22 M. Capinski, N. J. Cutland

By A we denote the self-adjoint extension of the operator -~ in IH and by {ek} an orthonormal basis of its eigenfunctions with the corresponding eigenvalues Ak, Ak ~ 0, Ak /" 00. For u E IH we write Uk = (u, ek). We write Prm for the projection of IH on the subspace IHm spanned by {el, ... , em}.

By V' we denote the space dual to V, with the duality extending the scalar product in IH.

In general, we define the spaces M for r > 0 by

00

M = {u E IH: LAkU~ < oo} k=l

(so IH = IFf, V = IHl). The elements of the dual spaces nrr (V' = nrl) are characterized by

00

lul-r := L AkrU~ < 00

k=l

where Uk = u[ek], so that

the limit being taken in nrr. The operator A can be extended to A : V -+ V' by Au(v) = ((u, v)) for v E V.

We put n r . 8vi .

b(u,v,z) = L ir ul(x)~(x)z'(x)dx = ((u, V)v,z) i,j=l D Xl

whenever the integrals make sense. Note that for u, v, z E V we have

b(u,v,z) = -b(u,z,v), (2)

hence b(u,v,v) = o.

We shall need some well-known inequalities for b (see [11] for example) and we list them here for reference:

Ib(u, v, z)1 < cilullllvllllzll, (3) Ib(u,v,z)1 < clulllvlllAzl, (4) Ib(u,v,z)1 < c lIulllvllAzl, (5) Ib(u,v,z)1 < c lui Ilvll Izlr if r > ~, (6)

for suitable u, v, and z.

Navier-Stokes equations

The inequality (3) allows us to define a V'-valued bilinear form B(u,v) by

B(u,v)(z) = b(u,v,z).

23

We now introduce a subspace of * IH which will playa crucial role in all our considerations. Fix an infinite integer N E *IN\IN and let IHN be the space spanned by the vectors {Eb ... , EN} where Ek = *ek.

By PrN we denote the orthogonal projection of * IH onto IHN. We use U, V for elements of IHN. We write Uk = (U, Ek) for k E *IN and we have

N N

(U,v) = L Uk Vk, 1U12 = Lui, k=1

N k=1

N

((U, V» = L AkUkVk, 11U112 = L AkUf. k=1 k=1

We shall also use the Ilf" norm of elements of IHN for r E JR (we shall mainly need r = 1,0,-1):

N

lUI; = L A'i,uf. k=1

In the following proposition we sum up some basic facts from nonstandard functional analysis that we shall need in what follows.

Proposition 2.1 For U E rnN and u E rnr , r E JR, we have

(a) U ~ u in the weak topology of rnr iff (U, *v) ~ (u, v) for all v E rn-r (and then Uk = aUk for all finite k);

(b) if U is nearstandard in rn r , then U is weakly nearstandard in rn r and the standard parts coincide.

We use st Ilf" U (w-st M U ) to denote the standard part (weak standard part) of U in Ilf". For U E IHN we write

00

°U = LOUkek, k=1

where we allow aUk = ±oo. In general this is a formal expression but if

00

L A'i,oU; < 00

k=1

for some r, then it makes sense and in fact ° U = w-st M U:

The next two lemmas are concerned with weak and strong nearstandardness, respectively.

24 M. Gapinski, N. J. Gutland

Lemma 2.2 Let r E JR and U E nIN • Then

(a) IOUlr $ °IUlr {we allow infinity on either side};

(b) if U is weakly IW -nearstandard, then °u = w-strnr(U);

(c) if IUlr < 00, then U is weakly rnr -nearstandard.

Lemma 2.3 For U E rnN and r E JR,

(a) U is {strongly} nearstandard in rnr iff

00 N

L::>-k Qui ~ L:>kUi < 00

k=l k=l

{we use x < 00 to mean IX is finite '};

(b) if IUlr < 00, then U is rnq-nearstandard for all q < r.

Remark Lemma 2.3 (b) is a nonstandard expression of the compactness of the embedding of IIr in JH'l for r > q.

The following crucial lemma has many applications.

Lemma 2.4 If IIUIl and IWII are finite with u = °U, v = °V, and z E domA then

"b(U, V, "z) ~ b( u, v, z).

Proof By transfer of the properties (4), (5) concerning b we have

I'b("u - U, "z, v)1 $ c I"u - UIIAzllI"vIL

l"b(U, "z, "v - V)I $ c I'v - VIIAzlllUli.

Since I"u - UI ~ 0 ~ I"v - VI and IAzl, 11U1I, and IIvll are finite, we readily obtain the result .•

3 Weak Solutions

We first give the definition of weak solution for the case of forces f depending on u. Such a general case is relevant since it allows us to deduce results about other equations of hydromechanics such as equations of thermohydraulics or magnetohydro-dynamics (cr. [12], where the linear dependence of f on u in two-dimensional case is dealt with).


Definition 3.1 Given Uo E DI and f : [0, T) x DI ~ V' the function u : [0,00) ~ DI is a weak solution of the N avier-Stokes equation if

(i) u E L2(0, T; V) n Loo(O, T; ill) for all T < 00,

(ii) for all v E V, for all t ~ ° (u(t),v) - (uo,v) = I t

[-v((u(s), v)) - b(u(s),u(s),v) + (f(s,u(s)),v)]ds (7)

where (f(s,u(s)),v) denotes the duality between V' and V extending the scalar product in DI

Theorem 3.2 Suppose that f is continuous and satisfies

If(t, U)I-1 ::; c(t)lIuli a luI1- a

for some a E [0,1), where c E V(O, T) for all T, where p ~ 1':a' Then there is a weak solution to Navier-Stokes equation.

Proof Consider an DIN valued function

N

U(r) = L Uk(r)Ek

k=1

and the following nonstandard equation in DIN:

k = 1, ... ,N, where r E *(0,00),

and the initial condition is

N

Uo = L(*uo, Ek)Ek •

k=1

By transfer we have local existence of a solution to (8). We shall show now that this solution exists for all r and satisfies a certain inequality. Multiplying both sides of (8) by Uk(r) and summing over k we find that

IU(aW + v 1" IIU(r)11 2dr = lUol 2 + 1" (F(r, U(r)), U(r))dr


(we have used the property b(u, v, v) = 0). Now, the right hand side can be estimated using the Young inequality ( ab ~ caP + cebq for p > 1, c > 0, where q = ~,

Ce = pSi;-l) and the hypothesis:

L7 (F(r,U(r)),U(r))dr < 1(7IF (r,U(r))I_dJU(r)lIdr

~ 1(7 c( r) IIU( r) 111+<> IU( r)) 11-<>dr

~ c 1(7 IIU(r)1I 2dr + Ce 1(7 cr:!a(r) IU(rWdr

where we require c < v. This inequality gives

which by Gronwall's lemma implies

where the right hand side is finite for finite a. The aim now is to show that U(r) has a standard part (in an appropriate sense) that is a weak solution of the N avier-Stokes equations.

By (9), for finite r, IU(r)1 < 00 so we have the weak standard part

w-stmU(r) = °U(r) E DI

with

for finite k.

Note that for finite T and each finite k, ·b(U(r), U(r), Er.) is S-integrable on ·[0, T) because using (4), and (9) we have

Clearly, the other terms on the right hand side of (8) are S-integrable. Hence if a I'=:j r < 00, °Ur.(a) = °Ur.(r) for all finite k and so U(a) I'=:j U(r) in the weak topology of DI (Le. U is S-continuous with respect to the weak topology of IH). Consequently, we can define a standard function

u: [0,00) ~ DI

by u(Or) = °U(r) and it is continuous with respect to the weak topology of DI. We claim that u is a weak solution to the Navier-Stokes equations.


Let T < 00. By Lemma 2.2 (a) for r = 0, and (9) we have u E Loo(O, T; IH). By Lemma 2.2 (a), r = 1, and basic Loeb theory we have

hence u E L2(0, T; V).

Finally it is sufficient to verify (7) for v = ek. For any t ~ 0 we have

(u(t),ek) = °Uk(t)

= (uo, ek) - °lt (vAkUk(r) + *b(U( r), U( r), *ek) - (* f( r), *ek)) dr.

Clearly

°It(*f(r),*ek)dr = 1tU(s),ek)ds;

and since Uk(r) is S-continuous we have

°lt AkUk(r)dr = 1t Akuk(Or)dLr = 1t AkUk(S)ds = 1t ((u(s), ek))ds.

Finally, * b(U( r), U( r), * ek) is S-integrable as noted earlier. Moreover, from (9) IIU(r)11 < 00 for a.a. r and so by Lemma 2.4

Thus

° 1t *b(U(r), U(r), *ek)dr = 1t o*b(U(r), U(r), *ek)dLr

= it b(u(Or),u(Or),ek)dLr = it b(u(s),u(s),ek)ds ° o.

which completes the proof. •

4 The Stochastic Case

We now turn to a general system of stochastic Navier-Stokes equations in dimensions n~4

{ du = [v~u- < u, V' > u + f(t, u) + V'pjdt + g(t, u)dWt (10) divu = 0

where W is an Dl-valued Wiener process of prescribed covariance Q (so Q E C(Dl, IH) is nonnegative, trace class), and the stochastic integral is a natural extension of the finite dimensional Ito integral.


Let us now introduce the following notations for the path spaces

ZT = Loo(O, Tj IH) n L2(0, Tj V) n G(O, Tj llIweak )

Z= n ZT T<oo

We can explain now what we mean by a solution of (10).

Definition 4.1 Let Uo E llI, f: [0,00) x V -+ V', and g: [0,00) x V -+ C(llI, V'). A stochastic process u( t, w) with a.a. paths in Z is a weak solution of the stochastic Navier-Stokes equations (10) if

E (sup lu(tW + f IIU(t) 1I2dt) :::; c(T) < 00 for all T (11) tE[O,T] 10

and

u(t) = Uo + It[-vAu(S) - B(u(s),u(s)) + f(s,u(s))Jds + It g(s,u(s))dw(s) (12)

holds as an identity in V' (the first integral is understood in the sense of Bochner).

Note that Vp as an element of Viis equal to 0: Vp[v] = (p, divv) = o. The properties of the two integrals involved mean that an equivalent formulation of (12) is that for each v E V

(u(t),v) = (uo,v) + I t [-v((u(s),v)) - b(u(s),u(s),v) + (f(s,u(s)),v)Jds

+ I t (v,g(s,u(s))dw(s)).

We first prove an existence theorem and then we discuss the possibility of relaxing some conditions on the coefficients. We prefer to give first the proof in a particular, though fairly ~eral, situation to avoid some technical complications which might obscure the simple underlying idea.

The Main Theorem

Let Km = {v : II v II :::; m} ~ V, which we consider with the strong topology of llI. In the theorem below, continuity on each Km turns out to be the appropriate condition for the coefficients f, g. Note that this is weaker than continuity on V in either the llI-norm or the weak topology of V.


Theorem 4.2 Suppose that Uo E IH and

f: [0,00) x V -+ V', 9 : [0,00) X V -+ C(IH, IH)

are jointly measurable functions with the following properties

(i) f(t,.) E C(Km' V~eak) for all m, (ii) g(t,.) E C(Km' C(IH, IH)weak) for all m,

(iii) If(t, U)I-l + Ig(t, u)IIH.IH :5 a(t)(l + luI) where a E L2(0, T) for all T.

Then equation (12) has a solution u on a filtered Loeb space, with (11) (hence u(·,w) E Z a.s.}.

Proof We will sketch the main thread of the proof, with reference to [4] for full details. Consider the following internal N-dimensional SDE for a stochastic process U(r,w) E IHN

{ dU(r) = [-v·AU(r) - B(U(r), U(r)) + F(r, U(r))]dr + G(r, U(r))dW(r) U(O) = PrN·uO

where, putting BN = PrN· B, the function B is given by

for some infinite K, and F, G are given by

F(r,U) = PrN·f(r,U),

G(r, U)V = PrN·g(r, U)V.

(13)

The growth and continuity conditions on f, 9 and the boundedness of B ensure (using the transfer of the standard theory of SDE's) that (13) has an internal solution U( r, w) for all r E ·[0,00) on some internal filtered space (0, A, (AT )T~O, II) carrying an internal Wiener process W with covariance QN; and U is adapted to (AT)T>o.

The transfer of Ito's formula gives

IU(rW + 2v loT IIU(O')11 2dO'

IU(OW + 2 loT (F(O', U(O')), U(O'))dO'

+ loT tr[G(O', U(O'))QNG(O', U(O'))']dO'

+2 loT (U(O'), G(O', U(O'))dW(O') (14)


Now

and

tr(G(r, U)QNG(r, U)') ~ trQNIG(r, U)llHN,lHN ~ trQN*a2(r}{1 + IUlf (16)

So, substituting (16), (15) in (14) we have, for any T

~~? IU(rW + v lT IIU(r)11 2dr

~ IU(O)12 + c lT *a2(r}{1 + lU(rW)dr + 2~~? IM(r)1 (17)

where M(r) is the internal martingale

M(r) = l r (U(a), G(a, U(a)))dW(a).

Now M has quadratic variation

where V(a) = (U(a), G(a, U(a))) E lHN.

Note that

So

[M](T) ~ csup IU(r)1 2 (*a2(r){1 + IU(rW)dr r5T Jo (18)

Now the Burkholder-Davis-Gundy inequality gives

for some fixed finite constant K. From (18) we have

and so


Substituting in (17) we obtain

and we can apply Gronwall's lemma with y(r) = IU(r)j2, g(r) = c3"a2(r), and h(r) = c(T)) to give EIU(r)j2:::; C4 for r :::; T. This implies

E (!~~ IU(rW + lT IIU(r)II2dr) :::; cl(T) (19)

with Cl (T) finite for T < 00.

Now put W(r,w) = G(r,U(r,w)). We have

IG(r, U(r, w)) I IHN,IHN :::; "a(r)(l + IU(r,w)1)

which together with (19) shows that

r f-? Iw(r,w)l~ IH N, N

is S-integrable for a.a. w, and

E (I T Iw(r)I~N,IHNdr) < E (IT "a2(r)(1 + lU(r)1)2dr)

< E (SUP(l + IU(r)l)2 t *a2(r)dr) T~T 10

< c2(T)

by (19). So by the Hilberl space extension of the theory of Ito integration on an adapted Loeb space (see [4], Theorem 5.1, for details)

foT G(a,U(a))dW(a)

is S-continuous in I . I for a.a. w.

By (19) again, B(U(r), U(r)) = BN(U(r), U(r)) for all finite r (for a.a. w), and so by Theorem 5.1 of [4]

IT [-v*AU(a) - B(U(a),U(a)) + F(a, U(a))]da

is a.s. S-continuous in V'. Hence U(r) is a.s. S-continuous in V' for finite r and we may define

u: [0,00) x n -+ IH


by u(t,w) = °U(r,w)

for any r ~ t. We will see that this u is the required solution. The estimate (19) gives condition (11) of the theorem.

To see that u satisfies equation (12), we have from Theorem 3.6 of [4]

° lot [-v • AU(a) - B(U(a), U(a)) + F(a, U(a))]da

= Iot[-VAU(S) - B(u(s),u(s)) + f(s,u(s))]ds. (20)

Finally, condition (iii) together with (19) shows that G(r,U(r)) is a.s. weakly nearstandard. Anderson's Luzin theorem shows that for a.a. finite r we have

all finite i, j, whenever 11U11 < 00 and so for a.a. w, for a.a. finite r

i.e. G(r, U(r,w)) lifts g(t,u(t,w)). Now by Theorem 5.4 of [4] have for a.a. w

o Iotc(a,U(a,w))dW(a) = lot g(s,u(s,w))dw(s) (21)

for all t < 00. Putting (20) and (21) together we see that u solves equation (12) .•

Remark We can easily adapt the proof to cover the case when Uo is random and independent of w. If Uo is given by a probability measure J], on DI then we take n = DIN X ·C(O, 00; DIN) and P = (J],N X AN )L, where J],N = • J], 0 PriV1 and AN is the internal Wiener measure.

Further Generalization

We replace condition (iii) of Theorem 4.2 by two separate conditions for f and g:

Cl) for a certain () E [0,1), and for a1 E Lq(O, T) with q = 1:9 we have

for all u E V.

C2) for a certain K, E [0,1), and for a2 E Lq(O,T) with q = 1~1t we have


for all u E V.

We claim that Theorem 4.2 remains valid if we replace (iii) by CI-2. The proof remains the same and we only present the necessary modifications.

First, using the Young inequality (a = 11U111+8v1¥-, b = al(r)IUI1- 8v-1¥-, P = 1!8) we have

to replace (15). Next, again by the Young inequality, we have

tr(G(r, U)QNG(r, Un < trQNIG(r, U)llHN,iHN < trQN"a~(r)11U112/tIUI2-2/t

v 1

< "211U112 + ca~-' (r)IUI2

which replaces (16). The resulting estimate (17) remains the same with a suitably chosen a.

Now, in the same fashion as above, we arrive at

which corresponds to (18). We can apply the Burkholder-Davis-Gundy inequality and we get

E (~~? IM(r)l) < KE (~~? IU(r)1 (IT "a~~' (r)IIU(r)1I 2/tIU(r)12- 2/tdr) !) < ~Esup IU(rW + ("ai~'(r)IIU(r)112/tIU(r)12-2/tdr

3 T~T Jo < ~Esup IU(rW + ~E rT IIU(r)112dr

3 T~T 2 Jo +cE lT "a~~·(r)IU(r)12dr.

Substituting in (17) we obtain

E (sup IU(rW + (1IU(r) 1I2dr) ::::; c5(T) + C6 E rT "a2(r)IU(rWdr T~T Jo Jo

and Gronwall's lemma gives (19) as before. In the proof of Theorem 4.2 we have not employed the full power of (14) as on the right of (16) and (18) we only had IU(r)12. Despite weaker estimates on F and G, (19) allows us now to draw the same conclusions about S-continuity of internal integrals and to complete the proof in the same way. At each step we use the Young inequality so that the terms involving IIU(r)1I2 on the right are accompanied by sufficiently small constants.


Statistical Solutions

First we give the definition of statistical solution for the stochastic system (10).

Definition 4.3 A family J.Lt of Borel probability measures on IH is called a statistical solution of the stochastic Navier-Stokes equations if

(i) the function t t-+ fIHIUI2dJ.Lt(u) is Loo(O, T) for all T < 00

(ii) lT fIHlluIl2dJ.Lt(u)dt < 00 for all T < 00

(iii) for any cp of the form cp(u) = ei(u,"l, v E V, and for any t > 0

fIHCP(u)dJ.Lt(u) = fIHCP(u)dJ.LO(u) + It fIH( -v((u, cp'(u))) - b(u,u, cp'(u))

+(f(s, u), cp'(u))

+~tr(QgT(s, u)cp"(u)g(s, u)) ) dJ.Ls(u)ds (22)

where Q is the covariance of w (gT denotes the transposed operator and cp', cp" are the functional derivatives of cp).

We shall call the equation (22) the Foias equation for the system (10). Comparing it with that derived by Foias [8] for the deterministic case it has an additional term involving the second order derivative cp" (u). The equation (22) is similar to that considered in [14] where g(t, u) = Id.

We now show that a solution of (22) is provided by the measures given by any solution to (10).

Theorem 4.4 Assume the hypotheses of Theorem 4.2. Suppose that the initialfunction uo E ill is random and independent from the Wiener process. Let J.L be the distribution of Uo and suppose that

Suppose that u(t,w) is an adapted solution to (10), satisfying (11), on a probability space (O,:F, P) with filtration (see Remark following Theorem 4.2).

Then the family of measures

J.Lt(A) = P(u(t) E A)

is a solution to the Foias equation (22), with J.Lo = J.L.


Proof Taking tp( u) = ei(u,v) with v E V, the infinite dimensional Ito lemmma in [10] gives

dtp(u(t)) = [-Il((u(t), tp'(u(t)))) - b(u(t), u(t), tp'(u(t)))

+(f(t, u(t)), tp'( u(t)) ) + ~tr( QgT (t, u(t) ) tp" ( u(t))g(t, u(t))) ]dt

+(tp'(u(t)),g(t,u(t))dw(t) ) = TJ(t, u(t))dt + 8(t, u(t))dw(t)

say. We shall prove below that

(a) E (IT ITJ(t,u(t))ldt) < 00 for all T < 00

(b) E (IT 18(t,u(t)Wdt) < 00 for all T < 00.

Taking (a) and (b) for granted, we can quickly complete the proof of the theorem:

JIHtp(u)d/-Lt(u) - JIHtp(U)d/-LO(U) = E(tp(u(t)) - tp(u(O)))

= E (It TJ(s, u(s))ds + lt 8(s, U(S))dW(S))

= lt ETJ(s,u(s))ds

using (a) and Fubini for the first term and the zero mean property of the stochastic integral given condition (b) ([10], Prop.1.4). Finally, note that

ETJ(s,u(s)) = JIHTJ(s,U)d/-Ls(U)

to see that (22) is satisfied. The conditions (i) and (ii) of Definition 4.3 follow from (11) automatically.

It remains to check (a) and (b) above. For (a) note first that IItp'(u)1I = IIvll and Itp"(u)IIH,IH= Ivl 2 so for any u E IH

1 ITJ(t, u)1 $ Illlullllvil + cllull 2 11 vll + If(t, u)lv' Ilvll + '2trQ.lg(t, u)Ik-,IHlvI2 •

The growth conditions (iii) on f and g, and (11) on lu(t)1 ensure that (a) holds.

For (b) it is enough to observe that

18(t,u)1 $ Itp'(u) I Ig(t, u)IlH,IH = Ivllg(t,u)IIH,IH

and use (iii) together with (11) again to see that

E (IT Ig(t,u(t))Ik-,mdt) < 00 ••


References

[1] A. Bensoussan, A model of stochastic differential equation in Hilbert space applicable to Navier-Stokes equation in dimension 2, in: Stochastic Analysis, Liber Amicorum for Moshe Zakai, eds. E.MayerWolf, E.Merzbach, and A.Schwartz, Academic Press 1991, pp.51-73.

[2] A.Bensoussan and R.Temam, Equations stochastiques du type Navier-Stokes, J. Functional Analysis 13 (1973), 195-222.

[3] M.Capinski and N.J.Cutland, Statistical solutions of Navier-Stokes equations by nonstandard densities, Mathematical Models and Methods in Applied Sciences 1:4 (1991), 447-460.

[4] M.Capinski and N.J.Cutland, Stochastic Navier-Stokes equations, Acta Applicanda Mathematicae 25 (1991), 59-85.

[5] M.Capinski and N.J.Cutland, A simple proof of existence of weak and statistical solutions of Navier-Stokes equations, Proceedings of the Royal Society, London, Ser.A, 436 (1992), I-II.

[6] M.Capinski and N.J.Cutland, Navier-Stokes equations with multiplicative noise, Nonlinearity 6 (1993), 71-77.

[7] M.Capinski and N.J.Cutland, Foias and Hopf statistical solutions for stochastic Navier-Stokes equations, to appear in Stochastics.

[8] C.Foias, Statistical study of Navier-Stokes equations I, Rend. Sem. Mat. Univ. Padova 48 (1973), 219-348

[9] H. Fujita Yashima, Equations de Navier-Stokes stochastiques non homogenes et applications, Thesis, Scuola Normale Superiore, Pisa 1992.

[10] A.lchikawa, Stability of semilinear stochastic evolution equations, Journal of Mathematical Analysis and Applications 90 (1982), 12-44.

[11] R.Temam, Navier-Stokes equations, North-Holland, Amsterdam 1979.

[12] R.Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York-Berlin-Heidelberg 1988.

[13] M. Viot, Solutions Faibles d'Equations aux Derivees Partielles non Lineaires, Thesis, Universite Paris VI (1976).

[14] M.I.Vishik and A.V.Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer, Dordrecht - London 1988.

HYPERFINITE ApPROXIMATIONS OF COMMUTATIVE TOPOLOGICAL GROUPS

E. 1. Gordon Department of Mathematics

University of Nizhny Novgorod Prospect Gagarina, 23

603600 NizhI)y N ovgorod, Russia

Abstract

In the paper [1] a nonstandard approach to the theory of Fourier series was based on the approximation ofthe group S = {z E <V : Izl = 1} by the hyperfinite group G = {z E <V : zN = 1} where N is an infinite large natural number. Here we prove the existence of such a hyperfinite approximation for an arbitrary LeA group with a compact open subgroup.

Introduction

In the paper [1] a nonstandard approach to the theory of Fourier series was developed. This approach was based on the approximation of the group S = {z E <Dlizi = 1} by the hyperfinite group G = {z E <DlzN = 1}, where N is an infinitely large natural number. In the paper [2-6] we considered the notion of a hyperfinite approximation (h.a.) for an arbitrary locally compact abelian (LeA) group, which made it possible to extend the ideas of the paper [1] to harmonic analysis on arbitrary separable LeA groups. The additive groups of the fields IR and ~ were considered in [2-6] as the examples. The existence of a h.a. for an arbitrary compact abelian group was proved in [5]. Here we prove the existence of a h.a. for an arbitrary LeA group with a compact open subgroup. This theorem was announced in [4]. We also present here some new examples of the hyperfinite approximations.

The nonstandard universe is assumed to be Wi - saturated. We use the following abbreviations:

Va (a standard :::} ... ) VA (A internal :::} ... )

(Similar for 3).

37

38 E. I. Gordon

1 The main theorem

In this paper g is a standard LeA group, *g its nonstandard extension, Mg the monad of zero in *g, Mg = n{ *UIO E U ~ g, U is open }; e g T/ := e'- T/ E Mg "Ie, T/ E *g (the indices g in the last two notations usually will be omitted); ns( *g) = {e E *gI3st T/ E g(e RJ T/)} : the homomorphism st : ns( *g) -+ g is defined by the condition st(e) RJ e. Definition 1. Let G be a hyperfinite group, and j : G -+ *g an internal map, Gf = rl(ns( *g)). The pair < G,j > is called a hyperfinite approximation (h.a.) of the group g if it satisfies the following conditions:

1) j(OG) = Og; 2) vste E g3g E G(j(g) RJ e); 3) "1gb g2 E Gf(j(gl ± g2 RJ j(gl) ± j(g2).

Definition 2. Let g be the dual group to g, and 0 the internal dual group to G. A h.a. < 0,3 > of g is called dual to a h.a. < G,j > of g if

1) "IX E O(XIGf RJ 1 :::} 3(x) RJ 0) 2) "IX E Of Vg E Gf(}(X)(j(g)) RJ X(g)).

Here Of = 3-1(ns(*g)), 0 Egis the zero of g, so "Ie E gO(e) = 1.

Let now g be separable, < G,j > - a h.a. of g and U ~ g a relatively compact neighbourhood of zero in g. Let

(1)

Define an internal measure I/c. on G such that VA C G(I/c.(A) = .::l·IA!), and let Ac. be the corresponding Loeb measure. The definition of I/f:" and Af:" on 0 is analogous. We consider G and g (0 and g) as the measurable spaces with a-algebras of Loeb measurable sets and Haar measurable sets respectively.

Theorem 1. i) The map st 0 j : Gf -+ g(st 03 : Of -+ g) is measurable and the Loeb measure Ac. (.xf:,,) on Gf (Of) induces the Haar measure /-Lc. (/-Lf:,,) on g (g).

ii) If f : g -+ <V is standard, then f E L1(g) iff there exist an S-integrable lifting F : G -+ * <V satisfying the following condition:

\fntA C G\Gf (.::l. ~ IF(g)1 RJ 0). (2) gEA

In this case Jg fd/-Lc. = O(.::l 0 2:gEG F(g)). Certainly the same theorem is true for

f E L1(g).

If standard p ~ 1, f E Lp(g) and F : G -+* <V is a lifting of f . st . j such that IFIP is S-integrable and satisfies (2) then F is called an Sp-integrable lifting of f.

Hyperfinite approximations of groups 39

Theorem 1 was proved in [4J, where we also investigated the conditions for * f 0 j to be an Sp-interable lifting of f E Lp(g). In the case of groups IR and <Q" the condition (2) is equivalent to some standard condition of fast enough decreasing of f at infinity [2.4J. This theorem has a standard interpretation, which deals with the approximations of Haar inegral on LOA groups by Riemann sums.

Let F: L2(g) --+ L2(g) be the Fourier transform on g :

and 'Ii : G'e --+ G'e the internal Fourier transform on G:

'Ii(¢)(~) = ~ E ¢(g)"e(g) gEG

In this case ¢-l(1f;)(g) = LiEeEt1f;(~)(g), where Li is defined by formula (1).

Theorem 2. If g has an open compact subgroup, and if {< G,j >, < G,] >} is the pair of dual h.a. of the groups g and g respectively then

i} the Haar measure /-Ll:" on g is associated with the Haar measure /-Ltl on g (this means that the Fourier transform F preserves normes, defined by these measures);

ii} if f E L2(g) and F : G --+* <D is an S2-integrable lifting of f, then ¢(I) is an S2-integrable lifting of F(I).

This theorem was proved in [6J. In [2,4J we had constructed the pair of dual h.a. of IR and R ~ 1R, which satisfies i) and ii) of theorem 2. Here we are going to prove.

Theorem 3. If g has an open compact subgroup, then g has a h.a. < G,j >.

Now, using theorem 4.1 of [4], it is easy to show that in this case g has a h.a., dual to < G,j >.

It is well known that every LOA group g is isomorphic to IRn x H, where H has an open compact subgroup. So, the following statement is true.

Corollary If g is a separable LeA group, then a pair of dual h. a. of g and g exists, which satisfies i} and ii} of theorem 2.

It is easy to present the standard version of this corollary, which establishes that the Fourier transform on arbitrary separable LOA group may be approximated by the Fourier transform on some finite group.

40 E. I. Gordon

2 The proof of the main theorem

We shall use the following well known van Kampen's theorem:

If g is a compact abelian group then in every open neighbourhood of zero it can be found such a closed subgroup H of g, that g/ H is isomorphic to sm x K for some natural m, and some finite group K.

Let U be a compact open subgroup of g, D = g/U. D is a countable discrete group (we assume that g is separable). Using wI-saturation of the nonstandard universe we can find a hyperfinite set T ~ * D such that D ~ T. Let * D(T) be the internal subgroup of * D, generated by T. If 7r : g --+ D is the canonical projection, H = *7r-l(*D(T)), c = *7rJH, then we have the following commutative diagram

U ~ ...

D g ----+ id,j.. id ..j.. id ..j.. (3) *U C H €

*D(T), ----+

the lines of which are short exact sequences.

Using the structural theorem about the finitely generated abelian groups and the transfer principle we find that * D(T) = Dl EB D2 where Dl is a hyperfinite abelian group and D2 an internal free abelian group with a hyperfinite set of generators. Let Hi = c-1(Di) (i = 1,2). Then H = HI + H2, HI n H2 = *U and the following sequences are exact

(4)

(5)

Here Ci = cJHi (i = 1,2).

Consider the infinitesimal neighbourhood of zero V ~ *U (so V ~ M). Using van Kampen's theorem and the transfer principle we find a hypernatural number k, a hyperfinite group R and a surjective continuous homomorphism ¢: *U --+ Sk EB R, such that Kercp ~ V. Using the extension theory for abelian groups we find such an internal group L, and such homomorphisms'Y : HI --+ L, '" :* Sk EB R --+ L, 0 : L --+ Db that the diagram

*U ~ HI ¢..j.. 'Y..j.. ~ cl

*Sk EB R ~ Ll ~ Dl (6)

is commutative, its lower line is exact and 'Y is surjective.

By the equality Ext(Db SkEBR) = Ext(Db R) there exist two short exact sequences

*sk "'1 L .It D ~ 1 -=-t 1, (7)


(8)

which are connected with the lower line of diagram (6) in following way:

The group *Sk is injective. So there exist a monomorphism x: D1 -* L1, such that 01' X = id. The group L2 is hyperfinite, because the groups Rand D1 are hyperfinite. Note that 1> is open, because it is a surjective continuous homomorphism of compact groups. So 1>(V) is an open neighbourhood of zero in Sk $ R, and 1>(V) n* Sk is a neighbourhood of zero in *Sk. It is easy to see that there is a hyperfinite subgroup F ~* Sk such that

(9)

Consider now the hyperfinite group M ~ L such that

M = {< K-1(f) + X(d), 1 > If E F, dE D1, l E L2, 02(1) = d}. (10)

The homomorphism, is surjective. So we can find an internal set G1 = {gmlm E M} such that Vm E M ,(gm) = m.

Define the binary operation +1 on G1 by the formula gml +1 gm2 = gml +m2' so that ,(gml + gm2) = ,(gml) +1 ,(gm2)' It is easy to see that < G1, +1 > is a hyperfinite abelian group.

Lemma 1. i) Vm1, m2 E M gml +1 gm2 ~ gml + gm2

ii) Vm E M (-lgm) ~ -gm

The proof follows immediately from the diagram (6).

Lemma 2. Vh E H13g E G1(g ~ h).

Proof Let hE H1. By the previous considerations ,(h) =< K-1(S) + X(d), 1 > where s E *Sk, l E L2, 02(1) = d. By (9) find f E F such that s - f E 1>(V) n* Sk. Then m =< K-1(f) + X(d), 1 >E M. It is easy to see that gm ~ h.

Let Gu =* un G1.

Lemma 3. Gu is a subgroup of G1• The pair < Gu, c > is a h.a. of U.

The proof follows immediately from the lemmas 1, 2 and the fact that U is a compact and open subgroup of g.

Let us consider now the sequence (5). let v: * D(T) -* D. (i = 1,2) be the canonical projection, and let m be hypernatural such that (v2(T) - v2(T)) n mD2 = {o}. To prove the existence of such m it is necessary to apply the transfer principle to the following proposition which is evidently true:

If P is a finite subset of a free finitely generated abelian group H then there is a natural m such that P n mH = {o}.

42 E.1. Gordon

Let Q = D2/mD2 and oX : D2 -+ Q the canonical projection. Then Q is a hyperfinite abelian group and oX is injective on v2(T). Let r : Q -+ D2 be such a right inverse map to oX that v2(T) ~ r(Q). Let G3 = r(Q) and Vq E Q r(q) = dq. Define the binary operation +3 on G3 by the formula dq1 +3 d'12 = dq1 +q2 (note that T is not a homomorphism!) so that oX(dq1 +3 dq2 ) = oX(dq1 ) + oX(dq2 ).

Lemma 4. i) vstd ED 3q E Q(v2(d) = dq);

ii) if dqp d'12 E v2(D) then dq1 +3 d'12 = dq1 + d'12

The operation + at the right side of the last equality is the group operation on D2.

We omit the simple proof of this lemma. Note only that by our construction D C T C *D, so v2(D) c v2(T) and v2(D) is an external subgroup of D2.

As D2 is a free abelian group it is projective. So there is a monomorphism J-L : D2 -+ H2 (5), which is right inverse to e2. Consider the set G2 = {g + J-L(dq)lg E Gu, q E Q} and define the operation +2 on G2 by the formular (gl +J-L(dq1 )) +2 (g2+J-L(dq2 )) = gl +u g2 + J-L(dq1 +3 dq2 )·

Using the facts that H2 = *U E9 J-L2(U2) and Gu C *U it is easy to see that < G2, +2> is a hyperfinite abelian group and G2 n* U = Gu.

Let now G = G1 X G2 and define j : G -+ *g by the equality j ( < gb g2 » = gl + g2. Let us show that < G, j > is a h.a. of g. The first condition of the definition 1 is obvious. Let e E g. As g C H = HI + H2 (3), so e = hI + h2' where h; E H;. Then d = 7r(e) = el(h1)+e2(h2) by (3), (4), (5) and e2(h2) = v2(d) E v2(D) because dE D. By lemma 4 v2(d) = dq for some q E Q and there is an element 9 E *U such that h2 = 9 + J-L(dq). By lemma 3 Gu approximates U and U is compact. So there is such go E Gu that go Ri g. Then g2 = go + J-L(dq ) Ri 9 + J-L(dq ) = h2 and g2 E G2. By lemma 2 there is gl E G1 such that gl Ri hI. Then gl + g2 Ri hI + h2 = e. We proved the second condition of the definition 1.

Let gl + g2 Ri e E g, g~ + g~ Ri' e' E g, g;,g: E G;(i = 1,2). We must show that (gl +1 gD + (g2 +2 g~) Ri e + e'. By lemma 1 (gl +1 gD Ri gl + g~. So it is enough to show that g2 +2 g~ Ri g2 + g~. Let 7r(e) = d. As gl + g2 Ri e so gl + g2 - e E *U and by (3) e(gl + g2) = d. By the definitions g; E H;, so e2(g2) = v2(d) E v2(D). Similar e2(g~) = v2(d') E v2(D), where d' = 7r(e'). Now by the definition of G2 g2 = 9 + J-L(v2(d)), g~ = 9' + J-L(v2(d')). By lemma 4 g2 +2 g~ = 9 +u 9' + J-L(v2(d) + v2(d')) 9 +u 9' + J-L(v2(d)) + J-L(v2(d')). By lemma 3 9 +u 9' Ri 9 + 9' and so g2 +2 g~ Ri g2 + g~. The considerations for gl - g2 are similar.

3 Some examples

Now we are going to present some examples of h.a. Let {< Kn,4>n > I n E IN} be a standard sequence of finite commutative abelian groups (rings) Kn and surjective homomorphisms 4> .. : Kn+l -+ K", K = lim+- < K", 4> .. > its projective limit.


By the definition of projective limits there is a sequence of surjective homomorphisms 4'?n : K --+ Kn such that ¢>n 0 4'?n+l = ¢>n' Let N E 'IN \ IN, then • KN is a hyperfinite abelian group (ring). Let j: 'KN --+ 'K be a right inverse map to ·4'?N. Then it is easy to prove the following

Proposition 1. < 'KN,j> is a h.a. 01 the group (ring) K. 01 course in the case 01 the rings we have to add to the definition 1 the condition 3) 'It gb g2 E G / j (a· b) ~ j(a)· j(b).

For example, if l' = {an In E N} is a standard sequence of natural numbers, such that 'ltn E IN anlan+l' Kn = lL./anlL., ¢>n : Kn+l --+ Kn is defined by formula ¢>n(a) = rem(a,an) - the remainder of division a by an,a E Kn+l (we present Kn as {O, 1,2, ... , an - I}), then K = ~T is the ring of 1'-adic integer numbers. The elements of K are the sequences {bnln E IN} of natural numbers, such that o ~ bn < an - 1, bn = rem(bn+b an). We can define the injection v : lL. --+ K such that 'ltb E lL. v(b) = {rem(b, an}lv E IN}. It is easy to see that lL. is dense in K. The following proposition follows immediately from Proposition 1.

Proposition 2. II N E 'IN \ IN, • KN = {O, 1,2, ... " aN - I}, j = vi • KN then < 'KN,j> is a h.a. 01 the ring ~T'

Let us construct now a dual h.a. to the h.a. of Proposition 2.

Let ~(T) = {:::'Im E lL.}. Then ~(T) is a subgroup of ~ because anlam for n ~ m and

lL. C ~(T). It is well known (see for example [7]) that ~(T) /lL. ~ AT' The character Xmn of ~Tl corresponding to the class m, n of the element.!!!. E ~(T) satisfies the , an

formula

Xm,n(() = exp21!'i m~:({), (E ~T' (11)

By transfer principle 'kN ~ ·KN.

Proposition 3. II J :' KN --+ AT is such that 'ltm E' KNJ(m) = Xm;n, then <' KN,J > is a dual h.a. to the h.a. of Proposition 2.

To prove the proposition 3 note that ~T is compact and in the case of compact groups the first condition of the definition 2 is always true [6J. The proof of the second condition is trivial by (11).

In the case of a compact group g if g --+ <V is bounded and almost everywhere continuous then '1 0 j is an Sp-integrable lifting of I for every p ;::: 1 (see the remarks after theorem 1). Then theorem 2 gives us the following approximation theorem for the Fourier transform on ~T'

Proposition 4. II I : ~T --+ <V is almost everywhere continuous then

lim L I I(() exp -21!'i-n -d/-LT(() - a~l L I(m) exp(- 1!'t m 12 = 0 an-1 1 (k1/; (() n-l 2 'k )

n-+oo k=O I::J. T an m=O an

Here /-LT is the normed Haar measure on ~T(/-LT(~T) = 1).

44 E. I. Gordon

The following example is the 7'-adic solenoid 2:,.. The set of this group is [0,1) x ~,.. The operation +,. is defined by formula

< x, e > +,. < y, 'f/ >=< {x + y}, e + 'f/ + [x + y] >,

where {x} is the fractional part and [xl-integer part of x. The definition of the compact topology on 2:,. can be found in [7]. We describe it only in the nonstandard language:

Propositon 5. Let N E* IN\IN, and let G be the additive group of the ring

·71../*a't·71.. = {a, 1, ... , a't -I}.

If j : G --+ .~,. is such that 'Va E G j(a} =< {a:}' [a:] >, then < G,j > is a

h.a. of:E,..

We omit an elementary proof of this proposition. Theorem 1 gives now the following approximation theorem for the Haar integral on :E,.. Proposition 6. If J.L is the normed Haar measure on :E,. and f : :E,. --+ a; is a bounded and almost everywhere continuous function on:E,. then

It is well known [7) that ~,. ~ <Q("). If 0: = .!!!. E <Q(") then an

21l'i(x + '1jJn(~}}m 'Vx E [0, 1] 'V~ E ~,. Xa(x,~} = exp .

an

Now it is easy to construct the dual h.a. As above G ~ G.

Proposition 7. If N, G,j are such as in Proposition 5 and] : G --+ f:,. is defined by the formula ](b) = XfJ. where f3 = a:' bEG, then < G,] > is a h.a. off:,., dual to < G,j >. Now it is quite easy to write the approximation theorem for the Fourier transformation on :E,. similar to Proposition 4.

If 7' = {pn In E IN} for a standard prime p, then ~,. = 71..p - the ring of integer p-adic numbers. If 7' = {n!ln E IN} then ~,. 9::! IIp 71p and f:,. 9::! (Q - the additive group of rationals in the discrete topology. So the Proposition 7 gives a h.a. for (Q.

Hypernnite approximations of groups 45

References

[1] Luxemburg, W.A.J.: '''A nonstandard analysis approach to Fourier analysis, contributions to nonstandards analysis'" , Amsterdam, North Holland 1972, p. 16 - 39.

[2] Gordon, E.!.: "'On Fourier transform in nonstandard analysis"', Izv. Vyssh. Uchebn. Zaved. Math. 1989, N 2 p. 17 - 25 (in Russian).

[3] Gordon, E.!.: "'Hyperfinite approximations of locally compact abelian groups"', Soviet Math. Dokl. 1991, Vol. 42, N 2 p. 567 - 571.

[4] Gordon, E.!.: '''Nonstandard analysis and compact abelian groups"', Siberian Math. J. 1991, Vol 32. N 2 p. 26 - 40.

[5] Gordon E.J.: '''Nonstandard analysis and locally compact abelian groups"', Acta Applicandae Mathematicae 1991, Vol. 25 p. 221 - 239.

[6] Gordon, E.!.: "'Hyperfinite approximations of locally compact groups and some of their applications"', to appear in the Proceedings of the 5-th Siberian School '''Algebra and Analysis"', Irkutsk, 1991.

[7] Hewitt, E.j Ross, K.: "'Abstract Harmonic Analysis"', 1963, Vol. 1, Springer - Verlag, Berlin-Gottingen-Heidelberg.

A NOTE ON THE MYOPE TOPOLOGY

Tommy Norberg t Dep. of Mathematics, CTH & GU, S-412 96 Goteborg, Sweden

Abstract

The note shows first that the myope and the Lawson topologies on the collection K of compact subsets of some locally compact Hausdorff space S are the same. Then, generalizing S to be some sober locally compact Q.s space and topologizing S = S u {~}, where ~ rt S, with the topology having closed sets !F = K U {S}, where K now denotes the collection of compact saturated subsets of S, it is shown that Fell's topology on !F relativizes to the myope (which by definition is the Lawson) topology on K. In other words, !F is a compactification of K. A characterization of the (nonstandard) monad of K E K w.r.t. the .myope topology is also given. Let mcc(s) denote the monad of s E S w.r.t. the cocompact topology. Then L E *K is near standard with standard part

st(L) = {s E S : mcc(s) n L #- 0}

if, and only if, L ~ * K for some K E K

1 Introd uction

The myope topology is a topology on the collection IC of compact subsets of some locally compact Hausdorff space S. It is generated by the two families

ICF := {K E IC : K n F = 0}, FE:F

ICc := {K E IC : K n G #- 0}, G E 9

(:F and 9 denote resp. the collections of closed and open subsets of S). It is the relativization to I( of the Vietoris topology topology on :F (see, e.g., [14]). Unless S is compact, the myope topology is strictly finer than the trace of Fell's 'hit-or-miss' topology on 1(, the latter being generated by the two collections I(K, K E 1(, and I(c, G E g. See Matheron [6], who shows Fell's topology to be compact Hausdorff and the myope topology to be locally compact Hausdorff.

tSupported by the Swedish Natural Science Research Council

46

On the myope topology 47

Our main aim with this short note is to give a nonstandard characterization of the monads of the myope topology. This has been done already for Hausdorff-S. Refer to Wicks [14, Propositions 2.1 and 3.1] and the references therein. We are, however, also interested in certain non-Hausdorff settings. Moreover, the manner in which we arrive to the characterization is interesting also for those who wish to restrict their horizon to Hausdorff spaces.

We begin by showing that the second subbase family above consisting of sets of the form /CG, can be thinned to G:s of the form G = S \ K with K E /C. This makes it possible to achieve our first goal, which is to extend the notion of the myope topology to certain non-Hausdorff settings in such a way that it is still being locally compact Hausdorff.

Its one-point compactification turns out to be homeomorphic to the collection j of all closed subsets of a sober locally compact space S. The topology on j is Fell's [1], known to be compact Hausdorff. Moreover, its monads are known [8], so seeing the myope topology as the trace on /C of Fell's topology tells us at once how its monads look like.

We proceed with a somewhat more detailed description of our results. The first, which is Theorem 1, states that the myope topology coincides with the Lawson topology on /C (cf. [2, 3]). This is not at all surprising, since also the latter is known to be a locally compact Hausdorff topology on /C and intuitively there cannot exist many topologies on /C with such strong regularity properties.

We then increase the generality of our setting by allowing S to be an arbitrary sober locally compact Q.s space, and redefining /C to be the collection of all compact saturated subsets of S. (For our readers convenience we recall in Section 2 some notions from (non-Hausdorff) topology that we believe are not widely known; let us just note at this point that Hausdorff spaces are sober and Q.s, and that all subsets of a Hausdorff space are saturated.) Relying on Theorem 1, we may (and will below) refer to the Lawson topology on /C as the myope. It is well known that this is a locally compact Hausdorff topology on /C, see, e.g., [2, 3].

Next a new space S is formed by adding to S a point ~ rJ. S. The topology we provide S with is not the usual one used in the one-point compactification of locally compact Hausdorff spaces. Instead the weaker one with non-empty open sets S\K, K E /C, is used. Its trace on S is the cocompact topology, cf. [3]. The topology we provided S with is sober and locally compact (Proposition 2 and Corollary 3). Thus Fell's topology on its collection j of closed sets is compact Hausdorff [1]. Our second characterization of the myope topology is Theorem 4, which states that the relativization of Fell's topology to /C coincides with the myope topology.

By seeing the myope topology as the trace of Fell's topology we obtain with hardly any work Theorem 5, saying that L E */C is near standard if, and only if, L ~ * K for some K E /C, and that in this case its standard part is

st(L} = {s E S: mcc(s} n L '" 0}

48 T. Norberg

(mcc{s) is the monad of s E 8 w.r.t. the co compact topology). By specializing to the case of a Hausdorff-8, we obtain in Theorem 7 the not surprising fact that a near standard L E */C has the standard part

st{L) = {s E 8: m{s) n L:f 0} = et: tEL}

(here m{s) and °t denote, resp., the monad of s E 8 and the standard part of tEL ~ *8 w.r.t. the original topology on 8; note that it follows from the fact that L is near standard that all points of L are near standard).

Besides giving a more thorough understanding of the myope topology, this note lays a ground for a forthcoming study of capacities using nonstandard analysis. In particular the kind of capacities that are associated with probability distributions or more general measures on the collection :F of closed subsets of 8, cf. [6, 8].

Any such capacity may be viewed as an increasing upper semicontinuous function from /C into the non-negative extended reals that maps 0 on 0 [6]. Writing C for the collection of capacities, let us just note here that any (internal capacity) c E *C has a standard part °c E C w.r.t. Vervaat's sup vague topology, since this is a compact Hausdorff topology [12], and that it follows by [9] that

(Oc)(K) = S-lim c{L) L~.K

for K E /C. (Refer, e.g., to [11] for a definition of S-right limits of internal functions on the hyperreals. Its extension to our setting is straightforward.) In order to calculate the S-limit on the right hand side above, we need Theorems 5 and 7 giving necessary and sufficient conditions (in the general and the Hausdorff case, resp.) for L E */C to be in the monad of K E /C.

The results are presented in detail in Section 2. Proofs that do not contribute significantly to the discussion are postponed to SectioIi 3.

2 Results

We begin by assuming 8 locally compact Hausdorff. The object of study is the myope topology on its collection /C of compact subsets. It is well known that /C is an upper continuous lattice, which among other things means that it carries the Lawson topology (refer to Gierz, Hofmann, Keimel, Lawson, Mislove & Scott [2] or Hofmann & Mislove [3]; note that these references treat the dual notion of lower continuous lattices; also: their notion of a continuous lattice is slightly more restrictive than ours, because they require of a continuous lattice that it is also a complete lattice, while we call a lattice continuous if it is continuous as a poset, cf. [5, 2, 3]).

Theorem 1 Let 8 be a locally compact Hausdorff space. On its collection /C of compact subsets, the myope and the Lawson topologies coincide.


The proof is postponed to Section 3. Let us only note at this point that the Lawson topology is generated by the two collections Kf, FE F, and I(Kc, K E 1(, cf. [3) (we write Be := S \ B). Thus, trivially, the myope topology is finer than the Lawson topology.

We now interupt the general development and recall some general topological notions. Let X be a topological space, and take B ~ X. We write B- and BO for its closure and interior, resp. Moreover, B is called saturated if B coincides with the intersection of its open neigborhoods, i.e.,

B = n{ G ~ X : G is open and contains B} =: satB

A simple exercise yields that x E satB if, and only if, {x} - n B f. 0. It follows, e.g., that all subsets of a T 1 space are saturated.

Note that we call X locally compact if every point x E X has a neighborhood basis of compact sets. This is not the standard definition of local compactness. It is, however, the definition used by Fell [1). Note that K ~ X is compact if, and only if, satK is so. Thus, in studying local compactness, restricting attention to compact saturated sets is no severe loss of generality. Hofmann & Mislove [3) claim that 'it is in fact advantageous to do so.' We do not disagree.

A closed set F ~ X is said to be irreducible if, whenever F ~ Fl U F2 for any closed sets Fb F2 ~ X, then either F ~ Fl or F ~ F2. It is clear that each singleton closure is irreducible. Now X is called sober if every non-empty irreducible closed subset is the singleton closure of a unique point in X, i.e., the mapping x t-+ {x} - is a bijection between X and its collection of non-empty irreducible closed sets. See [2, 3). Sober spaces are To: given any pair of distinct points, one has a neighborhood which does not contain the other; and a To space is called super-sober if the set of limit points of each ultrafilter is either empty or a singleton closure. All super-sober spaces are sober [2).

Note now that if both K, L ~ X are compact saturated, then so is K U L but not necessarily K n L. For an example of this phenomenon, see [10). Next, let {Ki}; be a collection of compact saturated sets in X. If X is sober, and the collection {Ki}; filtered, then ni Ki is compact saturated [3) ({ Ki}; is called filtered if for each pair Kill Ki2 there is some Kia ~ Kit n Ki2 ). By a Qc5 space we will understand a space in which the intersection of any non-empty collection of compact saturated sets is again compact saturated. If X is super-sober, then X is Qc5' A proof due to van Rooij of this fact may be found in [10) (see also [3]). Let us also note here that a sober locally compact Qc5 space is super-sober [3). So, for sober locally compact spaces, Q.s (actually the fact that K n L is compact, if both K, L so are) is equivalent to super-sobriety.

Returning to the general development, let S be a sober locally compact Q.s (equivalently, a super-sober locally compact) space, and write I( for its collection of compact saturated subsets. In the following we will refer to the topology on I(

50 T. Norberg

which is generated by the two families

J(:,F, F E:F, and ICKc, K E IC,

of sets in IC, as the myope topology. It is also known as the Lawson topology ( cf. [2, 3]), where it is shown that it is locally compact Hausdorff.

Let Ll be an ideal point not in S and consider the disjoint union

S:=SU{Ll}

We topologize S by declaring the sets S \ K, K E IC, open. Relativized to S, this topology is known as the cocompact topology [2, 3J. The set S endowed with the cocompact topology is denoted Sec. Note that Sec has a continuous topology. Its collection of closed subsets is the upper continuous lattice :Fcc := IC U is}. Everything would be fine if we could prove that Sec is sober, since then [3, Proposition 2.11J would allow us to conclude that Sec is locally compact. However, whether Sec is sober or not depends critically on whether S E IC or not. If S E IC then :Fcc = IC and we may conclude by [3, Lemma 2.23], that Sec is sober (to see how, refer to the proof below of Proposition 2). If S ~ IC then a simple argument yields that S itself is irreducible. This time [3, Lemma 2.23J tells us that S is not a singleton closure. Hence Sec is not sober in this case. In order to obtain a sober space provided with a continuous topology, we must form the larger S, the closed sets of which are the sets in j = IC U is}.

Proposition 2 Let S be a sober locally compact Q.s space, and let S = S U {Ll}, where Ll (j. S, be topologized by the sets in the collection {S \ K : K E IC} U {0}. Then S is sober.

For a proof, see Section 3. As noted already the topology of S is continuous, so the next result follows by [3, Proposition 2.11J.

Corollary 3 S is locally compact.

We now provide j with Fell's topology, which is known to be compact Hausdorff [lJ. Recall that it is generated by the two families

jK:= {F E j: F n K = 0}, K E iC

jG := {F E j: F n G f 0}, G E g of basic open sets (g and iC denote resp. the collections of open and compact saturated subsets of S). Being upper continuous, j also carries the Lawson topology, which is known to coincide with Fell's topology [2J.

Also the proof of the next result is deferred (see Section 3).


Theorem 4 Let S be a sober locally compact Q& space and write /C for its collection of compact saturated sets. Then the trace on /C of Fell's topology on j = /C U {8} is the myope topology. In other words, j is an one-point compactification of /C.

Next, we turn to a characterization of the monad m(K) of K E /C w.r.t. the myope topology. We work in a nonstandard enlargement of sufficiently high saturation (refer, e.g., to Hurd & Loeb [4]; our notations and terminology are standard). Recall that, for an arbitrary topological space X, the set

m(x) = n{*G: x E G, G open in X}

is called the monad of x EX. Moreover, y E * X is near standard if y E m( x) for some x E X. If distinct monads are disjoint, which they are, if, and only if, X is Hausdorff, then x E X is called the standard part of y E *X whenever y E m(x). This is written x = 0y or x = st(y).

Norberg [9) (cf. also [7, 13, 14); however, these references treat the case when S is Hausdorff only) characterizes the monad iii(F) of F E j w.r.t. Fell's topology: Let H E *j, then H is near standard with standard part

st(H) = if := {s E 8: iii(s) n H # 0} E j

(iii(s) is the monad of s E 8). Hence, for FE j, H E iii(F) if, and only if, if = F. By Theorem 4,

m(K) = iii(K) n */C

showing that if L E * /C, then L is near standard if, and only if, L E /C which of course holds true if, and only if, iii ( ~) n L = 0.

Almost all statements of the following theorem are now proved.

Theorem 5 Let S be a sober locally compact Q& space, and consider its collection /C of compact saturated sets provided with the myope topology. Let L E */C. Then L is near standard if, and only if, L ~ * K for some K E /C. In this case, the standard part of L is.

st(L) = {s E S: mcc(s) n L # 0}

Proof We first prove the assertion that iii(~) n L = 0 if, and only if, L ~ * K for some K E /C. To see it, note first that

iii(~) = n *8\ *K KEIC

It follows that iii(~) n L = 0 if, and only if,

L~ U *K KEIC

52 T. Norberg

By saturation, this holds true if, and only if, L ~ * K for some K E K,.

Next note that if L E *K" then L ~ *S. And noting that

mee(S) = m(s) n *S, s E S

we finally see that m(s) n L #- 0 if, and only if, mee(S) n L #- 0. • Note that, for s E S,

m(s) = n *8\ *K .~KEIC

Hence m(Ll) ~ m(s). Moreover,

m(Ll) := m(Ll) n *S

may be regarded as the monad of infinity. So a member of *K, is near standard if, and only if, it contains no infinite points. The next result cannot hold unless S is Hausdorff.

Proposition 6 Suppose S is a locally compact Hausdorff space, and let Sec be S endowed with the cocompact topology. Write m(s) and mee(s), resp., for the monads of s E S w.r.t. the original and the cocompact topology. Then

mee(s) = m(s) U m(Ll)

Moreover, the two sets on the right hand side are disjoint.

Our final theorem is not surprising in the light of Matheron's result saying that the myope topology coincides with the relativization of Fell's topology on subsets of K, that are compact w.r.t. the myope topology [6]. Being an obvious consequence of Proposition 6, it needs no further comment apart, perhaps, from the fact that if L E *K, then, by Robinson's characterization of compactness [4, Proposition III. 1. 12], every point of L is near a standard point in S. It can moreover be seen as a straightforward consequence of [14, Propositions 1.4 and 2.1].

Theorem 7 Let S be a locally compact Hausdorff space, and consider its collection K, of compact subsets in the myope topology. Suppose L E *K, is near standard. Then its standard part is

st(L) = {s E S: m(s) n L #- 0} = et : tEL}


3 Remaining proofs

Proof of Theorem 1 We have already noted that the myope topology is finer than the Lawson topology. To see the converse, let F E F and take Gb ... ,G,. E g. Suppose

K E K.g1, ... ,Gn := {H E K. : H n F = 0, H n Gi ¥ 0, ... ,H n G,. ¥ 0}

By local compactness of S, there is some L E K. satisfying K ~ LO ~ L ~ Fe. Then Li := G{ n L E K.. Moreover, K n Loe = 0 and K n Lie = K n Gi ¥ 0. Hence

K K. LoC E L1C, ... ,LnC

Suppose now that H K. LoC E L1C, ... ,LnC

Then H n Loe = 0, so H n F = 0. Moreover, H n Gi = H n Lie ¥ 0 since H ~ L. Hence

completing the proof. • Proof of Proposition 2 Let 0 ¥ I E j be irreducible. We may assume I E K. since S = {~} -. By [3, Lemma 2.23), I = sat { s} for some s E S. To see that this point is unique, it is enough to note that sat { s} = sat {t} if, and only if, {t} - = {sr. •

I;...roof of T':.,eorem (As alreadx noted Fell's topol?gy is generated by the two families F L , L E K., and FG, G E g. We know that S is locally compact (Corollary 3). Hence Fell's topology is compact Hausdorff [1). We may take G ¥ 0 in the latter subbase family Then G = S \ K for some K E K., so

JG = {H E j: H n S \ K ¥ 0}

Hence !fG n K. = {H E K. : H n S \ K ¥ 0} = K.Kc

(where the complementation is w.r.t. S). This takes care of the second family in the generator of Fell's topology, and now we switch attention to the first.

Let L E K. Then JL is a Scott open ideal in J (Le., JL is a lower set: if H E JL and H' E j,~H' ~ H, then also H' ~ jL; which is directed: if HiJ...H2 E JL then Hi U H2 E FL; and Scott open: if nt Ht E F L, where {Hth ~ F, then already ni Hti E JL for some finitely many HtU' .. ,Htn}. That JL is an ideal (Le., a lower directed set) is trivial. The fact that it is Scott open follows by standard compactness arguments.

Now if S E JL then JL = J, so we may assume S rt. JL. Then JL ~ K.. Thus jL is a Scott open ideal in K.. By [3, Corollary 2.17J and the Lawson duality [5, 2, 3), there is a unique G E 9 such that

JL = {K E K. : K ~ G}

54 T. Norberg

Clearly F := S \ G E F, and jL = Kf

This completes the proof. • Proof of Proposition 6 Note first that m( s) ~ mcc( s), since S \ KEg for all K E /C. Next let r E mcc(s) \ m(s). Suppose r E *K for some K E /C. Then r E m(t) for some t E K by Robinson's characterization of compactness, cr. [4]. Clearly t t- s. Thus we may choose K E /C such that t E KO while s E KC. Then, by the former, r E *Ko ~ *K and, by the latter, r E *S\ *K. This is clearly impossible, so r ¢. *K for all K E /C. Hence r E m(~), and we have proved the first inclusion in

mcc(s) \ m(s) ~ m(~) ~ mcc(s) \ m(s)

The remaining inclusion and the last statement of the proposition are trivial. •

Acknowledgement

The content of Theorem 1 grew out in discussions with Adrian Baddeley. I am moreover grateful to Keith Wicks for some helping comments to an earlier version of the manuscript.

References

[1] Fell, J. M. G.: A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Procedings of the American Mathematical Society 13, 1962, 472-476.

[2] Gierz, G., Hofmann, K. H., Keimel, K. Lawson, J. D., Mislove, M. & Scott, D. S.: A Compendium of Continuous Lattices. Springer, 1980.

[3] Hofmann, Karl H. & Mislove, Michael W.: Local compactness and continuous lattices. In Continuous Lattices (Pmc. Bremen 1979) (eds. Banaschewski, B. & Hoffmann, R.-E.), Springer LNM 871, 1981, 209-248.

[4] Hurd, Albert E. & Loeb, Peter A.: An Introduction to Nonstandard Real Analysis. Academic Press, 1985.

[5] Lawson, Jimmie D.: The duality of continuous posets. Houston Journal of Mathematics 5, 1979, 357-386.

[6] Matheron, G.: Random Sets and Integral Geometry. Wiley, 1975.

[7] Narens, Louis: Topologies of closed subsets. Transactions of the Americal Mathematical Society 174, 1972, 55-76.


[8] Norberg, Tommy: Existence theorems for measures on continuous posets, with applications to random set theory. Mathematica Scandinavica 64, 1989, 15-51.

[9] Norberg, Tommy: On Vervaat's sup vague topology. Arkiv for matematik 28, 1990, 139-144.

[10] Norberg, Tommy & Vervaat, Wim: Capacities on non-Hausdorff spaces. Report No 1989-11, Department of Mathematics, CTH & GU, 1989.

[11] Stroyan, K. D. & Bayod, J. M.: Foundations of Infinitesimal Stochastic Analysis. North-Holland, 1986.

[12] Vervaat, Wim: Random upper semicontinuous functions and extremal processes. Report MS-R8801, qWI Amsterdam, 1988.

[13] Wattenberg, Frank: Topologies on the set of closed subsets. Pacific Journal of Mathematics 68, 1977, 537-551.

[14] Wicks, Keith R.: Fractals and Hyperspaces. Springer LNM 1492, 1991.

NONLINEAR THEORIES OF GENERALIZED FUNCTIONS

MichaelOberguggenberger Institut fur Mathematik und Geometrie, Universitat Innsbruck,

A-6020 Innsbruck, Austria

Abstract

The purpose of this article is to present an overview of a branch of nonlinear analysis that has taken up rapid development during the past decade: the theory of "multiplication of distributions" or better the "nonlinear theory of generalized functions", as it goes far beyond the former topic by now. It has a significant overlap with nonstandard analysis; but within the boundaries of its scope, it takes a more general standpoint. We are interested in situations that involve (a) nonlinear operations; (b) differentiation; (c) singular objects (like measures or discontinuous functions).

1 Introd uction

The purpose of this article is to present an overview of a branch of nonlinear analysis that has taken up rapid development during the past decade: the theory of "multiplication of distributions" or better the "nonlinear theory of generalized functions", as it goes far beyond the former topic by now. It has a significant overlap with nonstandard analysis; but within the boundaries of its scope, it takes a more general standpoint. We are interested in situations that involve (a) nonlinear operations; (b) differentiation; ( c) singular objects (like measures or discontinuous functions).

The linear theory of distributions we take as a basis for our investigations, building upon the foundations laid down by L.Schwartz [34J using duality methods, but also keeping an eye on the sequential approach due to J.Mikusinski [19; 2J and G.Temple [36J. Distribution theory can successfully handle (b) and (c) simultaneously, while classical nonlinear analysis deals with (a) and (b), but predominantly avoids (c). It was realized in the early fifties (L.Schwartz [35J, H.Konig [13]) that the combination of (a) and (b) with (c) entails that the differentiation operators and the nonlinear

56

Nonlinear theories of generalized functions 57

operations cannot be simultaneously consistent with their classical counterparts. We shall present a framework in which (a), (b), and (c) can be treated in full generality, with a wealth of applications to nonlinear partial differential equations, and in which the consistency problem can be understood and dealt with. This new way of thinking about generalized functions is chiefly due to E.E.Rosinger [26-31] and J.F.Colombeau [4-6,8]' whose approach parallels ideas from nonstandard analysis, most prominently the earlier constructions of D.Laugwitz and C.Schmieden [33,14,15]. Exactly this "convergence of ideas" and a comparison thereof will be our theme.

The core of the circle of ideas is putting sequences back into their proper, important place. One may say that generalized functions are sequences, or families, of smooth functions, viewed as members of larger algebraic entities. To explain this viewpoint, let us look at the familiar example of the Dirac measure 8(x) in JRn • In linear distribution theory, all its families of regularizations are crystallized in the single object 8(x), and this conceptual simplicity is at the basis of the success of distribution theory in linear analysis. At the same time, this neglect of the regularization process appears as one of the reasons for the failure of classical distribution theory in handling nonlinear operations. This is reflected in the language of physicists, where the meaning of, say, the square 82 ( x) is considered "ambiguous". In short, to establish a nonlinear theory of generalized functions it is necessary to keep more information on the regularization process. Thus there will be various objects in a nonlinear theory of generalized functions which correspond to the classical Dirac measure, each representing different nonlinear properties. In the most general situation, all families of smooth functions approximating the Dirac measure might be taken into account. However, factorization comes as an important tool in reducing this immense amount of information. As we shall see, the factorization can be done in various ways so as to study analytical properties, for example stability, or to produce desired logical structures, as is done in the ultrapower constructions of nonstandard analysis. Thus we can formulate more precisely now: Generalized functions are members of factor algebras (or factor spaces) of families of smooth functions.

In keeping our framework on such a general level we shall be able to better understand the structural constraints as well as options and to produce different nonlinear properties, for example, a variety of consistency results with classical operations. We should like to mention that even this framework can be profitably further extended: A nonlinear theory of generalized functions can be based on Dedekind order completion as well as uniform completion of spaces of smooth functions (joint work in progress with E.E.Rosinger [24]).

The plan of exposition is as follows: In Section 2 we discuss various problems which arise when imbedding the space of distributions into differential algebras. In Section 3, a number of important examples of differential algebras of generalized functions are presented, and basic properties are discussed. Section 4 serves to make a comparison between these algebras, while in Section 5 we sketch a few of the by now numerous applications. Let us emphasize an aspect frequently arising in the appli-

58 M.Oberguggenberger

cations: there is a general transfer process that produces generalized solutions out of classical solutions; in the case of ultrapowers it actually coincides with the transfer principle of nonstandard analysis.

2 Constraints on imbedding the distributions into algebras

Given an open subset 0 of lRn , V'(O) denotes the space of distributions on 0, the dual of the space V(O) of infinitely differentiable, compactly supported functions on 0; L~(O) is the algebra of locally bounded functions and Ck(O), 0 :::; k :::; 00,

the algebra of k-times continuously differentiable functions on O. We should like to construct an associative, commutative algebra (A(O), +, 0) with the following properties:

(i) there is a linear imbedding of V'(O) into A(O), and the constant function 1 is the unity in A(O);

(ii) there are n derivation operators 01, ... , an on A(O), that is, linear maps satisfying the Leibnitz rule;

(iii) OJ I V'(O) coincides with the usual partial derivative, j = 1, ... , n;

(iv) 0 I Ll.;'c(O) x L~(O) coincides with the usual product of functions.

As already alluded to in the introduction, this seemingly benign list of differential algebraic properties and consistency requirements cannot be satisfied simultaneously. In fact, if we demand (i) and (ii) in an associative, commutative algebra then (iii) and (iv) mutually exclude each other. To see this, consider the case 0 = lR and let H be the Heaviside function (H(x) = 1 for x > 0, H(x) = 0 for x < 0). By (iv) we have that Hm = H for all m E IN in A(lR) , while the Leibnitz rule together with associativity and commutativity gives

so 1

HoH = -oH m

for all mE IN

whence oH = 0 contradicting (iii), according to which we should have oH = o. A slightly more complicated example, due to L.Schwartz [35], shows similarly that o I Ck(O) X Ck(O) cannot coincide with the usual product for finite k E IN, or else (iii) will be contradicted. In conclusion, as soon as we insist on imbedding V'(O) into an associative, commutative differential algebra, the new operations, when restricted to their classical domains of definition, can never be equal to the corresponding


classical operations (failure of strong consistency). The result can nevertheless be infinitely close to the classical one, in a sense to be made precise below (possibility of weak consistency).

We point out that J.F.Colombeau [4] was the first to show that one can achieve (i), (ii), (iii) and

(v) 0 I COO(n) x COO(n) coincides with the usual product.

In the setting of associative, commutative differential algebras, this appears to be the best possible result concerning the differential-algebraic structure. If one is willing to give up associativity and commutativity, then (i) through (iv) can be achieved at the same time. The corresponding construction is due to H. Konig [11,12] and constitutes the earliest nonlinear theory of generalized functions in the literature. However, we shall remain in the associative, commutative setting here.

We close this section by noting that the difficulties with the product H8H, or H8, are similarly apparent in the approach using regularizations. In fact, let He and 8e be smooth functions such that He ~ H, 8e ~ 8 as c ~ 0, say in the sense of distributions. Assume that He(x) == 0 for x ~ -c, He(x) == 1 for x ~ c. If 8e has its support in (-00, -c), then He8e == OJ if it has its support in (c, 00), then He8e == 8e. Thus lime-+o He8e takes on the value 0 in the first, the value 8 in the second case. This example conspicuously shows how the nonlinear properties depend on the regularization process.

3 Construction of algebras of generalized functions

To begin with, we shall investigate algebras of generalized functions on lR"j the case of generalized functions on open sets n c lR" will be indicated later. As the starting point we take an infinite index set E and consider the power set

We shall refer to the elements (Ue)eEE of X(lRn) as "sequences" of smooth functions, by a slight abuse of language. In fact, let us actually take E as the interval (0,1) for the time being.

Given any sequence (<Pe)eEE of compactly supported smooth functions converging to the Dirac measure as c ~ 0, we have an imbedding of the space of distributions into X(lR"):

~: V'(lRn) ~ X(lR")

w ~ (w * <Pe)eEE (1)

60 M. Oberguggenberger

Here the star denotes convolution. For example, one could take CPe of the form CPe(x) = Cncp(x/e) with cP E D(JRR), J cp(x)dx = 1. With the operations defined componentwise, X(JRR) is an associative, commutative differential algebra, and

OJ(W * CPe) = (Ojw) * CPe, j = 1, ... , n

whence Ojt(W) = t(Ojw)

for W E D'(JRR). Thus the algebra X(JRn} satisfies properties (i), (ii), and (iii) of Section 2; but certainly (v) or even (iv) do not hold. We also have the standard copy imbedding

a: COO(JRR) -+ X(JRn}

f -+ (f}eEE

We note that a is a morphism of differential algebras, so condition (v) would follow from the requirement that t I COO(JRR) = a, which fails to obtain here.

Remark 1. The dependence of the imbedding t on the choice of CPe can be made to disappear by changing the index set to E = D(JRR} and taking

W -+ (w * cP )'I'EV(JRn )

as the imbedding of "D'(JRn) into X(JRR). This does not change any of the differentialalgebraic properties but would make D' (JRn ) "canonically imbedded".

From the rather crude algebra X(JRn) we shall now construct more refined algebras by means of factorization. The advantages of this procedure have been indicated in the Introduction and will also be seen below; for example, it might be possible to identify t(f} and a(f) for f E COO(JRn) to produce property (v). Thus we shall take subalgebras X. C X and ideals I C X. and pass to the factor algebra A = X.II. As a preliminary attempt, consider

{(Ue}eEE E X(JRR) : Ue converges in D'(JRn} as e -+ O}

{(Ue}eEE E X(JRn) : Ue -+ 0 in D'(JRn ) as e -+ O}.

With this choice we can realize D' (JRn ) as a factor space

But Y(JRn) is not an algebra, and .1(JRn) is not an ideal, in whatever subalgebra of X(JRn ) that contains it. This reflects the fact that it is hopeless to try to define the structure of a reasonable algebra on D'(JRn) itself. One might say that Y(JRn )

is too small (it does not even contain polynomials of distributions), while .1(JRn ) is too large: we have lost too much information by identifying the elements of .1(JRn )

with zero.

Before presenting the most important positive examples let us have a brief look at the case of open subsets of JRn.


Remark 2. If 0 is an open subset of IRn , we can similarly define

The space of compactly supported distributions £'(0) is imbedded by

~o: £'(0) -+ X (0)

w -+ (w * CPe)eEE

where (CPe )eEE C V(lRn) is some fixed delta-sequence as above. An ad hoc cut-off method allows to imbed V'(O) as well: Take Xe C V(O) such that Xe == 1 eventually for small c, on every compact subset of 0, and set

(2)

for w E V'(O). We note that there are natural restriction maps

for 0 1 C O2 open. With these restriction maps, the family X (0), 0 C IRn open, becomes a sheaf. However, ~ is not a sheaf morphism (of vector spaces), nor is it true that 8jL(w) = ~(8jw), nor is ~ I £'(0) equal with ~o. The remedy for this is localization, and this leads us to our first example for the factorization process.

Example 1. Here we identify sequences in X (0) that vanish eventually on every compact set, putting

and letting

I,oc(O) = {(Ue)eEE EX (0) : VK cO compact 37] > 0

such that Ue I K == 0 for 0 < c < 7]}

This algebra is in the spirit of the theory of C.Schmieden and D.Laugwitz [33,15], who based their foundation of infinitesimal analysis on the sequential approach and identification by means of the Frechet filter on E as above (see also the discussion in [23, Section 23] for how far one can get with this approach in multiplication of distributions). This algebra was later reintroduced by Yu.V.Egorov [9,10].

We note that (i) and (ii) obviously hold in £(0), while property (iii) for the imbedding ~, defined by (2), follows from the fact that

eventually for small c on every compact set. Also, ~ is a sheaf morphism. A simple algebraic argument [23, Example 21.6] yields an alternative imbedding for which (i), (ii), and (v) are true.


Example 2. This is the familiar ultrapower construction of the algebra of internal smooth functions ·COO(O}. Let U be a free ultrafilter on E containing all terminal sets (0,1]), 1] > 0, and set

Iu(O} = {(U"}"EE EX (O) : {c: E E : u" == O} E U}

Then

'COO(O} = X (O}/Iu(O).

We note here right away that 'COO(O} is not a sheaf (in the standard sense). For example, consider the open cover W = {( -m, m) : m E IN} of R and the nonstandard function f defined as the class of (1/1(. - :))"EE in 'COO(O), where 1/1 is smooth and has compact support. Then f I (-m, m) == 0 for every mE IN, but f is not zero in 'COO(R}, violating one of the sheaf properties. Of course, 'COO(R} is a *-sheaf: we have to test the sheaf axioms using 'W; the behavior of f at infinitely large values matters, and indeed f I (-v, v) is not zero for certain v E 'IN\IN. The lack of the sheaf property causes some trouble concerning (iii): While for 0 = RR the imbedding t defined by (1) commutes with derivatives, this is not the case for 0 f:. RR where we have to use formula (2). The contributions of 8j X" remain present as long as we do not factor out what happens off compact sets for small c: E E. Consequently, we can assert that 'COO(O} has property (iii) only when 0 = RR. Again, this difficulty can be removed by localization.

Example 3. Define

Iu,loc(O} = {(U"}"EE E X (O) : VK C 0 compact

it holds that {c: E E : Ue I K == O} E U}

and let

The family 'C/:c(O}, 0 C RR open, forms a sheaf in the standard sense, and each of its members is a differential algebra satifying (i), (ii), (iii). In discussion with T.D.Todorov we realized that these algebras can also be introduced in purely nonstandard fashion as factor algebras of ·COO(O}. Indeed, let

K(O} = {u E 'COO(O}: VK C 0 compact, U I'K == O} = {u E 'COO(O} : u(x} = 0 for all nearstandard x E ·O}.

A basic property of iterated factorization entails that

'C/:AO} = ·COO(O}/K(O}.

Algebras of this type are discussed in T.D.Todorov's paper [40]. We shall have a glimpse at more elaborate versions (in polysaturated models) in Remark 3, Section 4.


Example 4. This is one of the versions of the construction of J.F.Colombeau [5,6]. Let

and set

XM(11) = {(Ue)eEE E X (11) : VK C 11 compact, Va E INo 3p > 0, 3'/] > 0 such that

sup 18"ue(x) I::; e-P , 0 < e < '/]}, xEK

N(11) = {(Ue)eEE E X (11) : VK C 11 compact, Va E INo Vq> 0, 3'/] > 0 such that

sup 18"u,(x) I::; eq , 0 < e < '/]} xEK

Q (11) = XM(11)/N (11).

Apart from the fact that it seems natural from the standpoint of asymptotic analysis to consider the elements of N(11) negligible (this has been decidedly done e.g. by V.P.Maslov and V.A.Tsupin [17,18]), Q (11) and its variants currently are the only known algebras enjoying all of the properties (i), (ii), (iii), and (v). To see this, take cp E S (JRn ), a rapidly decreasing smooth function, with J cp(x)dx = 1 and all moments vanishing:

/ x"cp(x)dx = 0, for all a E INo, 1 a I~ 1.

The imbedding LO : EI(11) -+ Q (11),

W -+ class of (w * Cp" )eEE

with cp(x) = Cncp(x/e) has the property that LO 1 V(11) = 0', because (w*Cp" -W)eEE belongs to N(11), as follows from Taylor expansion. From there an easy exercise in sheaf theory gives L : VI(11) -+ Q (11) with L 1 COO (11) = 0'.

Example 5. This is a nonstandard version of Q (11); it was introduced in [20,23] and by T.D.Todorov [37,40]. Let

XM,u(11) = {(U,,)eEE E X (11) : VK C 11 compact, Va E INo 3p> 0 such that {e E E: sup 18"u,,(x)1 ::; e-P } E U},

xEK Nu (11) = {(U,,)eEE E X (11) : VK C 11 compact, Va E IN;)

Vq> 0 it holds that {e E E : sup 18"u,,(x)1 ::; eq } E U}, xEK

where U is a free ultrafilter on E as in Examples 2 and 3. Finally, set

Qu(11) = XM,u(11)/Nu(11).

We note that the corresponding ring of constants Qu(JRO) is a field, and actually coincides with the field PJR introduced by A.Robinson [25], where p is the fixed infinitesimal number given as the class of (e)eEE in *JR. See T.D. Todorov [40] for a further discussion of this correspondence. We remark that Qu(S1) can also be introduced as a factor algebra in *COO(11).


Example 6. This construction is due to E.E.Rosinger [29-31]. Define the "nowhere dense ideal" by

and set

Ind(O) = {(Ue)eeE E X (0) : 3r c 0 closed, nowhere dense, and

Vx E O\r 377 > 0 3V c O\r, V neighborhood of x,

such that U e 1 V == 0, 0 < c < 77}

R (0) = X (O)/Ind(O).

It is clear that R (0) is a differential algebra and that it contains C~(O), the algebra of classes of smooth functions defined off nowhere dense, closed subsets of O. To show that D'(O) is contained in R (0) requires a completely different line of argument; the imbeddings (1) and (2) will not work, as all delta-sequences (CPe)eeE with supports shrinking to {O} belong to Ind(IRn).

For further details as well as additional examples we refer to E.E.Rosinger [29-31] and [23, Section 21].

We close this section with a discussion of some generalities concerning all examples given. The ring of constants has already been mentioned; it can be introduced as the space of generalized functions on IRo = {O}. For example,

XM(IRO) = {(Ce)eeE E IRE : 3p> 0, 377 > 0 such that 1 Ce I:::; c-P, 0 < c < 77}

N(IRO) = {(Ce)eeE E IRE : Vq > 0, 377 > 0 such that 1 Ce I:::; cq , 0 < c < 77}

and Q(IRO) = XM(IRO)/N(IRO) is the ring of constants in Colombeau's theory. Similarly, *IR = X(IRO)/Iu(IRO) is the field of hyperreal numbers, and so on. All these rings are partially ordered in the obvious way (c ~ 0 iff it is the class of some nonnegative sequence (Ce)eeE). Further, IR is always contained as its standard copy, and one has infinitely large and infinitely small numbers.

Now let X.(O) be a subalgebra of X (0), I (0) an ideal therein, and A(O) = X.(O)/I(O) the factor algebra. Given 1jJ E D(O) and U E A(O), where U is the class of (ue )eeE, we can define the integral

f u(x)1jJ(x)dx = class of (f ue(x)1jJ(x)dX)eeE

as a member of the corresponding ring of constants A(IRO). With the aid of this integral, a relation of infinitesimality can be introduced in A(O). Indeed, given U E A(O) we shall write

U~o

provided

V1jJ E D(O), Vr E IR, r> 0 : If u(x)1jJ(x)dxl :::; r.

In the special case of *COO(O), this is the familiar relation of distributional infinitesimality, while in Q (0) it is the so-called association relation. The above relation can


be considered in all the Examples 1-5, but not in R (0) because there u E Ind(O) does not imply u ~ o.

As a preliminary application of this concept, we note that weak consistency of the product in A(O) with e.g. the product of continuous functions can be formulated. Indeed, if /,g are members of CO(O) then their product in A(O) is infinitely close to their classical product, i.e.

This is an immediate consequence of the weak convergence of (f * 'Po)(g * 'Po) to / g, where 'Po is the regularizing sequence appearing in the definition of ~.

We have claimed in the Introduction that there will be various objects with different nonlinear properties, each corresponding to the Dirac measure. This can now be made precise. Such an object 8 E A(JR) is meant to be infinitely close to 8(x), that is,

We illustrate the variety by noting that for every c E <D there is such a 8 E A (JR) which in addition satisfies

(see[23, Example 10.6]; complex-valued versions of our algebras A (JR) are needed here ).

Finally, we observe that the structure of the algebras A(O) generally goes much beyond that of differential algebras. For example, one can form nonlinear functions of the elements of A(O). More precisely, if F is an infinitely differentiable function on JR and u E A(O), then F(u) is a well-defined element of A(O) in the cases A(O) = £(0), ·C""(O), ·C!:c(O), 'R (0). In the cases A(O) = Q (0), Qu(O) this is also true, provided F is polynomially bounded together with all its derivatives, i.e. F E OM(JR). This observation is important for applications to partial differential equations with smooth, non-polynomial nonlinearities.

4 Comparison

In this section we are going to compare the algebras introduced in Examples 1-6 with respect to certain elementary properties. Let us start right away with a systematic


table. Numbers (i)-(v) refer to the properties defined in Section 2.

(i) (ii) (iii) (v) ring of superposition

differential oa I V' Coo 8ubal- constants by 8mooth F 8heaf

:)V' algebra consistent gebrs via t i8 field p088ible

e ye8 ye8 either-or? no ye8 ye8

·coo ye8 ye8 either-or?? ye8 ye8 no

*C~c ye8 ye8 either-or? ye8 yeB y ..

g ye8 yeB yeB y .. no only OM yeB

gu ye8 yeB yeB yeB ye8 only OM yeB

R yeB yeB no? ye8 no ye8 ye8

Proofs of these properties are either simple or else can be found e.g. in [23, Sections 9,21,23]. The "no" with question mark in the last line means that so far no imbedding of V'(O) into R(O) is known which produces property (iii); the "either-or" in lines 1-3 means that imbeddings are known which either produce (iii) or else (v). The one giving (iii) is the imbedding t discussed in Section 3, formula (1), (2). The algebraic argument of [23, Example 21.6], alluded to in Example 1 above, works also for ·Coo and ·C,~c and yields an alternative imbedding with property (v). The double question mark in line 2 refers to the fact that we can assert property (iii) for ·Coo(O) only when 0 = Rn. We shall elaborate the either-or situation in more detail in Remark 3 below.

Aside from the elementary properties listed in the table above, each of the algebras has various distinguishing, advanced properties. For example, ·Coo is part of the nonstandard universe, and so the fundamental methods of nonstandard analysis can be applied, like the transfer principle or saturation arguments, as we shall see below in Remark 3.

Next, g (0) appears to be the only algebra allowing for a remarkable regularity theory. Let us explain this with the notion of elliptic regularity. Take a constant coefficient partial differential operator P( 8) which is elliptic or, more generally, hypoelliptic. These operators possess the following classical regularity property: If wE V'(O) and P(8)w E Coo(O), then wE Coo(O). Such a result never holds in any of the algebras above: Not even U E A(O), P(8)u = 0 implies U E Coo(O), because we can take U as an element of the ring of constants minus the real numbers. However, in the setting of Colombeau's theory we can construct a substitute for Coo(O) to formulate regularity: Define goo(O) as the algebra of those elements U E g (0) all whose derivatives have the same local order, i.e. the representatives (Ue)eEE of U are required to satisfy that:

V K C 0 com pact 3p > 0 such that V Q E 1N~ 317 > 0 with sUPxEK I 8aue (x) 1:$ c-P , 0 < c < 17.

One can show [23, Section 25] that V'(O) n goo(O) = Coo(O), and one has the regularity result: If U E g (0) and P(8)u E goo(O), then U E goo(O).


A distinguishing feature of 'R (0) is the fact that it includes C~(O), as we have noted in Example 6. This is of interest in the theory of shock wave solutions to quasilinear hyperbolic systems. Shock waves are functions smooth off a finite set of smooth curves or surfaces and hence belong to C~(O). In fact, a quasilinear conservation law

(3)

can have shock wave solutions in the sense of equality in 'R (0), while in all the other algebras (3) has to be replaced by

OtU + oxF(u) ~ 0

in order to accommodate shock wave solutions. This difficulty comes from an interplay of the differential-algebraic structure, the Rankine-Hugoniot conditions for the jump of a shock wave, and the distributional convergence properties of the members of the ideal under consideration; details can be found in [23, Sections 19 and 22] and [22, Proposition 3.5].

Finally, a remarkable fact deserves attention: the sheaves A(O) = C (0), ·C~c(O), 'R (0) are flabby. That is, if 0 1 C O2 are open sets and U E A(Ol), then there exists v E A(02) such that v I 0 1 = u. Indeed, as the boundary 001 is closed, there is , E COO(02) such that 001 = {x E O2 : ,(x) = a}. Take X E COO(R), X == 0 on ( -1, 1), X == 1 on (-2, 2 f. If (uE )EEE is a representative of u, define v as the class of the sequence of functions

vE(x) = X (~,(x)) uE(x).

It is immediately verified that UE == VE on every compact subset of 0 1 for sufficiently small e, so the sequence (uE I 0 - VE )EEE belongs to any of the defining ideals for the algebras above, hence v I 0 1 = u. The sheaves Q (0) and Qu(O) are not flabby, because the order of growth e-P of U E can increase beyond bounds near OOb whereas V E should have bounded order on and near a01 •

Remark 3. The either-or situation with properties (iii) and (v) in the table above can be nicely demonstrated by means of ·COO(JRn). At the same time, this is a good opportunity to discuss what saturation can do for us. We choose the index set E so large and the ultrafilter appropriately so that our nonstandard universe (built on ·R, say) becomes polysaturated (see e.g. T.Lindstr!1Sm [16] for the terminology). The following argument we learned from H.Akiyama [1]. Due to polysaturation, there is a vector space 'Ii of hyperfinite dimension l/ such that

V(Rn) C 'Ii C ·V(JRn)

(see [16, Proposition 111.3.9]). Consider the inner product (cp,'l/J) -t f cp(x)¢(x)dx inherited from ·V(Rn) on 'Ii. Choose an orthonormal basis {'l/J1, ... , 'l/Jv} in 'Ii and define!:::" E ·V(JR2n) by

v

!:::"(x, y) = L'l/Jj(x)¢j(Y) j=l


Define an imbedding K, of V'(lRn) into ·coo(lRn) by assigning to w E V'(lRn) the nonstandard function

x -+ (·w,.6.(x,.))

where the brackets denote duality in ·V'(IRR), ·V(IRR). It is immediate to check that

J .6.(x,y) .cp(y) dy = ·cp(x) (4)

for all cp E V(IRR), X E ·IRR; hence K, I V(lRn) = (1, and this is almost property (v). A partition of unity argument shows that equation (4) holds also for cp E coo(lRn) at all finite x E ·IRR. Therefore, passing to ·C~C(IRR), the map K, defines an imbedding V'(IRR) -+ ·C~(lRn) with K, I COO(IRR) = (1. Thus we have property (v). However, it appears that nothing can be said about derivatives; we do not know whether K,(8j w) = 8j K,(w) for W E V'(lRn) (note that integration by parts does not work because 8x .6.(x, y) f= -8y .6.(x, y), otherwise .6. could not have *-compact support).

On the other hand, using a different saturation argument, T.D.Todorov [38,39] has constructed a nonstandard function 8 E ·V(lRn) with the property

J 8(x - y) .cp(y) dy = ·cp(x)

for all cp E CO(IRR) and standard x E IRR. The map>. : V'(IRR) -+ ·COO(IRR) which assigns to each W E V' (IRR) the nonstandard function ·w * 8 is an imbedding which commutes with derivatives and hence satisfies (iii). This time we cannot infer property (v), because (·cp*8)(x) = ·cp(x) is only known to hold for standard x E IRR, so .cp * 8 and .cp need not be equal as nonstandard functions.

5 Applications

We begin with a general observation. Let T( 8) be a possibly nonlinear differential operator with smooth coefficients on 0 C IRn. Given two factor algebras AI(O) = XI/Il, A 2(0) = X2/I2 with Xl, X2 C X(O), we can define

provided T(8), acting componentwise, maps Xl into X2 and

(5)

An element u = s + II of Al (0) is a solution to the equation T( 8)u = 0 iff


Thus Ii determines the amount that the representative s can be perturbed and still defines a solution, that is, the size of Ii corresponds to stability. On the other hand, the size of I2 corresponds to exactness, i.e. the degree how much T( 8)( s) is allowed to differ from zero and yet is considered a solution.

We note that (5) is valid if in particular T(8) is polynomially nonlinear and Ai(n) = A2(n) is one of the differential algebras listed in Section 3.

Let us now see how solutions effectively are constructed in a simple, illuminating example: the Cauchy problem for the semilinear wave equation with generalized functions as initial data. Denoting the coordinates of lR? by (t, x), we consider the problem

(8; - 8;)u + F(u) = 0 on lR2

u I {t = O} = a,8t u I {t = O} = bon lR. (6)

Suppose we are given a type of factor algebra A(lRn) = X.(lRn)/I(lRn), n = 1,2. For the purpose of this discussion, we take the same algebra on both sides of the differential equation, assuming that T(8) = 8; - 8; + F(.) maps A(lR2) into itself, so that any solution u E A(IR?) will satisfy the equation in the differential-algebraic sense of A(IR?), while the restriction to {t = O} is defined componentwise on representatives. We now put forth a list of three basic hypotheses, which roughly amount to requiring that problem (6) with classical COO-initial data has unique COO-solutions, and possibly admits certain COO-estimates; more precisely:

(HI) if a", b" E Coo(lR) then there is a unique solution U" E coo(lR?);

(H2) if (a")"EE, (b")"EE belong to X.(JR) then the solution sequence (U")"EE belongs to X.(JR2);

(H3) if (U")"EE and (V")"EE belong to X.(JR2) and

(T(8)ue - T(8)ve )eEE E I(JR2 ),

(u" - v" I {t = O} )"EE, (8t u" - 8t v" I {t = O} )"EE E I(JR)

then (u" - V")"EE E I(JR2).

As an immediate consequence of these hypotheses we have the following result:

Proposition: Let a, b E A (JR) and suppose that (Hl)-(H3) hold. Then problem (6) has a unique solution u E A(JR2).

Proof: Let (a")"EE, (b")"EE be representatives of the initial data a, b E A(lR). According to (HI) and (H2), the classical solution sequence (U")"EE can serve as a representative for a solution U E A(JR2). (H3) translates directly into uniqueness of generalized solutions .•

All what remains to do is to verify hypotheses (Hl)-(H3). Now (HI) is known to hold if F is smooth and e.g. F' is bounded. In the case X.(lRn) = X(lRn), (H2) is trivially true, while finite propagation speed for the wave equation and uniqueness


of classical solutions gives (H3) for I/oe, Iu, Iu,/oe of Section 3. Thus we have unique solutions in A(R2) = C(R2), *coo(R2), *C~e(R2). The case of Q(R2), QU(R2) requires some more work: here the estimates entering in XM and N have to be verified; these are classical a priori bounds for COO-solutions. Thus we also have unique solutions in Q(R2) and QU(R2).

Let us pause here for a moment and reflect upon what we have achieved: We have produced generalized solutions, almost without effort, by building up from classical COO-solutions, going to sequences, and then factoring. As pointed out in the Introduction, this can be viewed as a transfer process (and in the case of *coo(R2) it actually is nothing but the transfer principle at work).

Now, as V'(R) C A(R) we have in particular unique solutions to the nonlinear problem (6) when the initial data are arbitrary distributions. This, of course, goes much beyond what could be obtained or even formulated in the classical, distributional setting. However, in certain cases classical solutions do exist. For example, if a, b belong to CO(R) then there is a unique solution v E CO(R2). What is the relation of v to the generalized solution u E A(R2) with initial data t(a), t(b), where t is the imbedding of V' into A ? Well, we generally can assert that u is infinitely close to v, that is, u ~ t(v). This is just a translation of the fact that v depends continuously on the initial data. However, it is in general impossible to have u equal to t( v) in A(R2); this is caused by reasons similar to the negative results of Section 2. We do have u = t(v) in Q(R2) and QU(R2) provided a, b belong to COO(R), thanks to the property t I Coo = a. Finally, it is possible to achieve strong consistency u = t( v) for merely continuous a, b if one is willing to discard the differential algebraic structure and work in vector spaces instead (to be sure, different ones on both sides of the equation, and thus explointing more of the power of the general factorization framework). Such a construction is worked out in [21] and [23, Section 24], it involves factoring X(R2) by the space of zero-sequences in CO(R : Ltoc(R)).

We mention that so-called delta waves can be accommodated as well: These are solutions with initial data measures or distributions with discrete support. Here various cases are known where the classical, approximate solutions with regularized initial data converge weakly to a limit v E V'(R2), and then the generalized solution u E A(R2) is again infinitely close to v.

We now list a number of further applications that have been obtained during the past few years, or are in the process of being worked out. References will be omitted, but can be found in [23, Sections 16-20, and Sections 22,26,27], together with further details. We also ask the reader to look at the monographs of H.A.Biagioni [3], J-F.Colombeau [5,6,8] and E.E.Rosinger [28-31] as well as the survey article of J.F.Colombeau [7].

(1) The results above on the Cauchy problem with singular data are not confined to the wave equation; they hold more generally for semilinear hyperbolic systems and equations in one space dimension, and have also been extended to several space dimensions.


(2) It may happen that instead of the initial data some coefficients are singular: the same methods allow to handle hyperbolic equations with discontinuous coefficients, as e.g. arising in transmission problems in layered media.

(3) Quasilinear hyperbolic equations and systems. In the case of a conservation law

OtU + oxF(u) = 0

smooth approximations can be constructed from the parabolic perturbation

OtU + oxF(u) = /-LO~U

with /-L ~ 0; these approximate solutions then satisfy Otu+oxF(u) ~ 0, and thus one can study shock waves (as indicated in Section 4), singular data, and asymptotic behavior in terms of /-L. Our algebras provide also a setting for the study of shock waves in nonconservative systems, where the usual weak solution concept fails. There is the possibility to employ both equality and the infinitesimality relation in order to fix the jump conditions. A typical system that has been studied is

8t u + u8x u 8x v

OtV + uOxv ~ oxu.

(4) A number of special equations have been investigated: transport equations, like the Carleman system, the Korteweg - de Vries equation, Schrodinger equations, the p-system from gas dynamics, and so on.

(5) Regularized derivatives. Given a generalized function u on IRn with some representative (U~)~EE' the regularized derivatives 8j u, j = 1, ... , n can be defined by means of an additional convolution with a delta-sequence:

8j ue = OJ(ue * 'Pe)

for c E E. The Cauchy problem for an evolution equation in IRn +1 can be rewritten by replacing the space derivatives with regularized derivatives while keeping the usual time derivative:

ul{t=O}

L aa 8~u lal5m

Uo·

This has the effect that for each fixed c the original partial differential equation is turned into an ordinary integro-differential equation, which is readily solved. This way arbitrary evolution equations, irrespective of their type (be it hyperbolic, parabolic, ... ) have unique generalized solutions, backward and forward in time. This approach also offers the opportunity to construct unconditionally stable difference schemes (cf. E.E.Rosinger [32]).

(6) The theory of pseudodifferential operators can be transplanted into the setting of 9 (11) and used to prove regularity results involving gOO(11).


(7) The Cauchy problem for analytic partial differential equations (Cauchy-Kowalewski type), which generally can only be solved locally in the classical setting, admits global solutions in the algebra R(n). This relies on the fact that classical solutions can be constructed off a nowhere dense, closed set which carries the singularities.

(8) Quantum field theory. Each renormalization prescription determines an object in an algebra of generalized functions; the otherwise symbolic computations involved thus acquire a differential-algebraic meaning.

(9) Generalized stochastic processes can be accommodated by viewing their paths as generalized functions; this yields a reduction to smooth paths for fixed c E E.

This list of applications may suffice. We hope to have shown that going into the structure of factor algebras is a rewarding venture. On the one hand, it delivers a variety of generalized functions and supplies a wealth of new objects which can be used for precise modelling. On the other hand, it clarifies the issue of nonlinear operations with them and unifies various approaches, from weak solutions to nonstandard analysis. Finally, it provides a general framework for dealing with nonlinear differential equations and, by taking a fresh viewpoint, raises a range of new questions.

6 References [1] H. Akiyama: Applications of nonstandard analysis to diffusion on manifolds, this volume.

[2] P. Antosik, J. Mikusinski, R. Sikorski: Theory of Distributions. The Sequential Approach, Elsevier, Amsterdam 1973.

[3] H.A. Biagioni: A Nonlinear Theory of Generalized Functions, Lecture Notes Math. 1421, Springer, Berlin 1990.

[4] J.F. Colombeau: Une multiplication generale des distributions, C.R. Acad. Sci. Paris, Ser.l, 296 (1983), 357-360.

[5] J.F. Colombeau: New Generalized Functions and Multiplication of Distributions, North Holland, Amsterdam 1984.

[6] J.F. Colombeau: Elementary Introduction to New Generalized Functions, North Holland, Amsterdam 1985.

[7] J.F. Colombeau: Multiplication of distributions, Bull. Am. Math. Soc. 23 (1990),251-268.

[8] J.F. Colombeau: Multiplication of Distributions. A tool in mathematics, numerical engineering and theoretical physics, Lecture Notes Math. 1532, Springer, Berlin 1992.

[9] Yu. V. Egorov: On a new theory of generalized functions, (Russian), Vestnik Moskov. Univ. Ser.l, 1989, No.4, 96-99.


[10] Yu. V. Egorov: A contribution to the theory oj generalized Junctions, Russian Math. Surveys 45:5 (1990),1-49. Translated from: Uspekhi Mat. Nauk 45:5 (1990), 3-40.

[11] H. Konig: Neue Begriindung der Theorie der "Distributionen" von L. Schwartz, Math. Nachr. 9 (1953), 129-148.

[12] H. Konig: Multiplikation von Distributionen. I, Math. Ann. 128 (1955), 420-452.

[13] H. Konig: Multiplikation und VariablentransJormation in der Theorie der Distributionen, Arch. Math. 6 (1955), 391-396.

[14] D. Laugwitz: Eine EinJiihrung der 8-Funktionen, Sitzungsb. Bayer. Akad. Wiss., Math.-nat. Kl. 1959, 41-59.

[15] D. Laugwitz: Anwendung unendlich kleiner Zahlen. I. Zur Theorie der Distributionen, J. reine angew. Math. 207 (1961), 53-60.

[16] T. Lindstr~m: An invitation to nonstandard analysis, In: N. Cutland (Ed.), Nonstandard Analysis and its Applications,. Cambridge Univ. Press, Cambridge 1988, 1-105.

[17] V. P. Maslov, V. A. Tsupin: 8-shaped Sobolev generalized solutions oj quasilinear equations, Russion Math. Surveys 34:1 (1979), 231-232. Translated from: Uspekhi Mat. Nauk 34:1 (1979), 235-236.

[18] V. P. Maslov, V. A. Tsupin: Necessary conditions Jor the existence oj infinitesimally narrow solitons in gas dynamics, SOy. Phys. Dokl. 24 (5) (1979), 354-356. Translated from: Dokl. Akad. Nauk SSSR 246 (1979), 298-300.

[19] J. Mikusiriski: Sur la methode de generalisation de M. Laurent Schwartz sur la convergence Jaible, Fund. Math. 35 (1948), 235-239.

[20] M. Oberguggenberger: Products oj distributions: nonstandard methods, Zeitschrift Anal. Anw. 7 (1988), 347-365. Corrections to this article: Zeitschr. Anal. Anw. 10 (1991), 263-264.

[21] M. Oberguggenberger: Semilinear wave equations with rough initial data, In: P. Antosik, A. Kaminski (Eds.), Generalized Functions and Convergence,. Worid Scientific Publ., Singapore 1990, 181-203.

[22] M. Oberguggenberger: Case study oj a nonlinear, nonconservative, non-strictly hyperbolic system, Nonlinear Anal. 19 (1992), 53-79.

[23] M. Oberguggenberger: Multiplication oj Distributions and Applications to Partial Differential Equations, Pitman Research Notes Math. 259, Longman, Harlow 1992.

[24] M. Oberguggenberger, E. E. Rosinger: Solution oj Continuous Nonlinear PDEs through Order Completion, Preprint 1992/1993.

[25] A. Robinson: Function theory on some nonarchimedean fields, Am. Math. Monthly 80 (6), Part II: Papers in the Foundations of Mathematics (1973), 87-109.


[26] E. E. Rosinger: Scufundarea distribulitor V~ in algebre pseudotopologice, Stud. cerc. mat. 18 (1966), 687-729.

[27] E. E. Rosinger: Spatii pseudotopologice. Scufundarea distribulitor V~ in algebre, Stud. cerc. mat. 20 (1968), 553-582.

[28] E. E. Rosinger: Distributions and Nonlinear Partial Differential Equations, Lecture Notes Math. 684, Springer, Berlin 1978.

[29] E. E. Rosinger: Nonlinear Partial Differential Equations. Sequential and Weak Solutions, North Holland, Amsterdam 1980.

[30] E. E. Rosinger: Generalized Solutions of Nonlinear Partial Differential Equations, North Holland, Amsterdam 1987.

[31] E. E. Rosinger: Non-Linear Partial Differential Equations. An Algebraic View of Generalized Solutions, North Holland, Amsterdam 1990.

[32] E. E. Rosinger: Nonlinear Equivalence, Reduction of PDEs to ODEs and Fast Convergent Numerical Methods, Pitman Research Notes Math. 77, Pitman, Boston 1982.

[33] C. Schmieden, D. Laugwitz: Eine Erweiterung der Infinitesimalrechnung, Math. Zeitschr. 69 (1958], 1-39.

[34] L. Schwartz: Theorie des distributions, Nouvelle ed., Hermann, Paris 1966.

[35] L. Schwartz: Sur l'impossibilitf de la multiplication des distributions, C. R. Acad. Sci. Paris 239 (1954), 847-848.

[36] G. Temple: Theories and applications of generalized functions, J. London Math. Soc. 28 (1953), 134-148.

[37] T. D. Todorov: Colombeau's generalized functions and nonstandard analysis, In: B. Stankovic, E. Pap, S. Pilipovic, V. S. Vladimirov Eds.), Generalized Functions, Convergence Structures, and Their Applications. Plenum Press, New York 1988, 327-339.

[38] T. D. Todorov: A nonstandard delta function, Proc. Am. Math. Soc. 110 (1990), 1143-1144.

[39] T. D. Todorov: Pointwise kernels of Schwartz distributions, Proc. Am. Math. Soc. 114 (1992), 817-819.

[40] T. D. Todorov: Nonstandard asymptotic analysis and nonlinear theory of generalized functions, Preprint 1993

A NONSTANDARD ApPROACH TO THE PETTIS INTEGRAL

Horst Osswald Mathematisches Institut, UniversWit Miinchen

Theresienstr. 39, D-80333 Miinchen

Abstract

We shall prove lifting theorems for Pettis measurable stochastic processes and Pettis integrable functions on Loeb spaces with values in nonseparable locally convex spaces. We apply the results to give a nonstandard proof of

[1] Vitali's convergence theorem for uniformly Pettis integrable functions with values in weakly complete spaces.

We will show for functions with weakly compact range:

[2] the existence of conditional expectations in adabted Loeb spaces,

[3] Keisler's Fubini theorem,

[4] the existence of solutions to special cases of nondeterministic vector valued Peano-Caratheodory differential equations.

1 Introduction

It is well known that each Borel measurable function f on a finite Loeb space (0, L(A), fl) with values in a Banach space has a lifting (i. e., an internal representation) if and only if the range of f is separable valued jL-almost everywhere. Therefore Loeb-theory seems to be restricted only to functions with values in separable spaces.

By means of linear continuous functionals we introduce a notion of "lifting" in order to study functions with values in nonseparable spaces. Our procedure provides a nonstandard approach to Pettis-measurable functions and stochastic processes and to the Pettis integral.

It seems to be natural and is useful later to extend the notion of Pettis integrability (Pettis [15]) in the following way: Fix a topological vector space 18 (over IR or <V) and a separable Banach space ID such that there exists a separating set 18' (i. e., for each a E 18 , a =I- 0, there exists a cp E 18' with cp( a) =I- 0) of continuous linear operators from 18 into ID . A function f from a measure space (A, B, v) into 18 is

75

76 H. Osswald

now called Pettis integrable, if cp 0 f is Bochner integrable for all cp E IB' and if for every set B E B there exists a vector aB E IB such that for all cp E IB'

cp(aB) = B -l cp 0 fdv (in the sense of the Bochner integral).

Then aB is called the Pettis integral of f over B, denoted by P - IB fdv.

Among other things we will show that the Pettis integral of a Pettis integrable function f on a Loeb space can be written -up to an infinitesimal error- as the internal integral of a *finitely valued function F, where F is an appropriate lifting of f. Important is that F does not depend on cp E IB'. This result is a nonstandard version of a result due to R. F. Geitz [5], which (the nonstandard version) we obtain here in a comparatively simple way.

In the case when IB is a Banach space, ID = <V and IB' is the topological dual of IB the result shows in an intuitive manner the difference between Pettis- and Bochner integrability: (The result for the Bochner integral can be found in [12].)

A function f : n --t IB is Bochner (Pettis) integrable if and only if there exists an internal measurable *simple function

F : n --t *IB such that (a), (b) and (c) hold:

(a) IIF(w) - f(w)1I ~ 0 for fL-almost all wEn (11cp 0 F(w) - cp 0 f(w)11 ~ 0 for all cp E IB' and fL-almost all wEn).

Notice that fL-a.a. depends on cpo

(b) II· II 0 F is S -integrable. (cp 0 F is S-integrable for all cp E IB').

(c) IA FdJ-L is nearstandard in the norm (weak) topology for all A E A.

Whenever F is such a function, then

In FdJ-L ~ B - In fdfL in the norm topology Un FdJ-L ~ P - In fdfL in the weak topology).

Our lifting results can be used as a tool to prove vector valued versions of well known results (due to Anderson [2], Keisler [9] , Loeb [10]) in scalar valued Loeb theory.

Acknowledgement

I would like to thank Peter Loeb and Jerry Uhl for many stimulating comments.

2 Preliminaries

We will work with a polysaturated enlargement of a standard model M. We assume that IB, <V and ID are elements of M, so every subset oflB, <V and ID and every function on these sets with values in these sets and so on belong to M, so IB' E M.

A nonstandard approach to the Pettis integral 77

For undefined notions and for the basic of nonstandard analysis consult the book of S. Albeverio, J. E. Fenstad, R HliSegh-Krohn and T. LindstrliSm [1] or the book of A.E. Hurd, P.A. Loeb [8]. If there is no risk of confusion, we omit the left upper index *. For example, we will always write 11·11, cp, with cp E IB', instead of "11'11, "cp.

In order to fix the terminology, I will give a short and simple introduction to scalar valued Loeb spaces. The idea comes from P. Loeb's functional approach to nonstandard measure theory [11].

Let (0, A, IL) be an internal measure space such that IL : A ---+ "[O,oo[ is limited. An internal or external subset NCO is called a IL-nullset, if for a 11 standard E > 0 there exists A E A with N c A and IL(A) < E. Let N denote the set of alllL-nullsets. Using the saturation assumption, it is easy to check that the union of a sequence in AnN of length at most card(M) is a subset of an element of AnN. Moreover, note that N is closed under countable unions.

Let B C 0 be external or internal and let A E A. Then A is called aIL-approximation of B, if the symmetric difference A 6 B of A and B is a IL-nullset. It is easy to see that the set

L(A) := {B C OIB has aIL-approximation}

is a a-algebra and that p, is well defined and a-additive, where p,(B) := °IL(A)) for all B E L(A) and all IL-approximations A of B. (0, L(A), p,) is called the Loeb space over (0, A, IL) .

Whenever A = "P(O) = {A cOlA is internal }, then we will write L(O) instead of L(A).

3 Weak nonstandard hulls of m The notion "nonstandard hull" for topological vector spaces was developed by W. A. J. Luxemburg [13] and extensively studied by C. W. Henson and L. C. Moore. See for example [7]. We use weak nonstandard hulls in order to show that not only functions with values in IB, but also functions with values in the algebraic duallB'~ of IB' have "IB-valued liftings.

Fix a linear space IBo over CD between IB and the linear space s"IB, where

s"IB := {a E "IBlcp(a) is nearstandard in the norm topology on ID for all cp E IB'}.

Define for all a E "IB

a := {b E *IBlcp(a) ~ cp(b) in the norm topology on ID for all cp E IB'}.

An element a E "IB is called ftearstandard and a is called the standard part of a, if a n BanBo '" 0. Define

113 := {ala is nearstandard }.

78 H. Osswald

Notice that 113 depends on IBo. 113 is a linear space over <V with the natural definitions of + and· and (IB,IB/) is a dual pair with respect to the bilinear mapping (a, cp) I---t

°cp(a). Define for all nearstandard a E *IB

cp: a I---t °cp(a)

113 is called a weak nonstandard hull of lB. Often we identify a E IB with a E 113, because a I---t a is an embedding of IB into lB. We will write a ~w b or a ~w b, if a = b. For every subset L C ·IB we define

Examples 3.1 (a) Let 10 := <V and let IB' be the topological dual of lB. Notice that for all a E s·IB the function ~a : IB' ~ <V, defined by ~a(CP) = °cp(a), belongs to the algebraic duallB'~ of IB' and that for each ~ E IB/~ there exists an a E s·IB such that ~ = ~a. (See Henson and Moore [7].)

({3) 1flBo = IB then 113 = lB. Thus IB itself is a special example of a weak nonstandard hull of lB. Further examples for nonstandard characterizations of spaces between IB and IB'~ can be found in the paper of S. Heinrich [6}.

(,) Suppose that L is an internal set of nearstandard points. Then L is a weakly compact subset of 113. Because L is embeddable into IB/~, where IB'~ is here the set of all linear operators on IB' with values in 10 , one may copy the proof of W. A. J. Luxemburg's result [13] that the standard part of an internal set of nearstandard points in a regular space X (i. e. X is a Hausdorff space, in which every point has a neighbourhood base of closed sets) is compact.

For the remainder of this paper we fix a nonstandard hull 113 of IB .

4 Pettis measurable stochastic processes

In the following fix a Loeb probability space (A, L(A), M and let S be a Hausdorff'space with countable base (On)nelN and 10 a separable Banach space with countable base (Un)nelN. We denote the set of functions f from S into 10 with f[X] c Y by W(X, V). Let W(n,m) := W( On, Urn), where Urn is the closure of Um. We identify functions f : A X B ~ C with f', where f' : A ~ {gig : B ~ C} with f'(a)(b) = f(a,b). An internal function F: n ~ X is called A-measurable, if F has only *finitely many values and if for all a E X

{w I F(w) = a} E A.

The next result, which is an extension of Keisler's "uniform lifting theorem" for functions with values in locally compact spaces S (se e [9]) to a larger class of spaces, is due to D. Ross [16), where he used capacity arguments. Because his result


is important in the following and because we want to have statement (+), we give the proof, which uses arguments similar to the original proof of the Loeb- Anderson lifting theorem (see [9]) for random variables on a Loeb-space. The proof here is different from Ross' and Keisler's proof.

Lemma 4.1 Let f : n X S ---+ ID. Then the following statements (1) and (2) are equivalent:

(1) (a) f(·, s) : n ---+ ID is measurable for all s E Sand

(b) f(w,·) : S ---+ ID is continuous for [J,-almost all wEn.

(2) There exists an internal function F : n x ° S ---+ OlD such that

(d) F : n ---+ {G : * S ---+ *ID IG is internal} is A-measurable and

(b' ) for [J,-almost all w E A

F(w, s) Rl f(w, Os) for all nearstandard s E • S.

More precisely, we will prove for the implication "(1) ==:} (2) (fJ)" that for all n, m E IN and [J,-almost all wEn

(+) if f(w) E W(n,m), then F(w) E 'W(n,m)'

Proof: Because the implication "(2) ==:} (1)" is well known, we only prove the implication "(1) ==:} (2)". Let D be a countable dense subset of S. Since

rl[W(n,m)] = n U(',p) E Um} and U("p) E Urn} E L(A), peDnOn

f-I[W(n,m)l E L(A). It follows that there exists a J.L-approximation A(n,m) of rl[W(n,m)]' Let (IN x 1N,f-) be a well ordering of order type (IN,<). As in the Loeb-Anderson lifting theorem we modify A~ to Aa such that Aa remains a J.Lapproximation of f-I[Wal and for each finite sequence (al,"" ak) in IN x IN

(++) Aa, n ... n Aa. =I- 0 ==:} 'Wa, n ... n ·Wa• =I- 0.

To this end define by recursion

Aa := A~ \ U{Aa, n ... n Aa.lal f- ... f- ak f- a and Wa, n ... n Wa• n Wa = 0}.

Using (++) and a saturation argument we obtain an A-measurable function

F : A ---+ {G : • S ---+ olD I G internal }

such that for all a E IN x IN F[Aal C *Wa. It follows that for all w rf. UaelNxlN AaLV-I[Wal such that f(w,') is continuous and all {J E IN x IN

f(w) E Wf3 <=> wE r l [Wf3] <=> wE Af3 ==:} F(w) E 'Wf3 ,

thus f(w, Os) Rl F(w,s) for all nearstandard s E ·S .•

80 H. Osswald

Lemma 4.2 Let L C *18 be internal. For each mapping h : S ---+ t there exists an internal mapping H : * S ---+ L such that lor all cp E 18' and all (n, m) E IN X IN

rp 0 hE W(n,m) ~ cp 0 HE *W(On' G) lor all open G ::> Urn

Proof: Set X:= {(cp,n,m)lrpoh E W(n,m)}'

Because for each finite sequence (CPl, nb mt}, ... ,(cpk, nk, mk) in X and open Gl ::> Umll ••• ,Gk ::> Umk

by saturation, there exists an internal function H : * S --+ L such that for all (cp,n,m) E X and all open G::> Urn:

H[*Onl C cp-l[*G], i.e., cp 0 HE *W(On' G) .•

Before we give the main result in this section, we define I : n X S --+ 18 to be Pettis measurable, if for all cp E 18'

(1) for J1-almost all wEn, where J1-a. a. depends on cp (!), rp 0 I ( w, .) : S --+ ID is continuous and

(2) rp 0 1(', s) : n ---+ ID is measurable for all s E S.

An A-measurable function F : n ---+ {G : * S ---+ LIG internal} (then, by definition, F has *finitely many values) is called a weak lifting of I: n x S --+ t, if for all cp E 18' and J1-almost all wEn (again fJ,-a. a. depends an cp) and all nearstandard a E *S

cp 0 F(w, a) ~ rp 0 I(w, °a).

The next result is an extension of the Keisler Ross lifting theorem for continuous stochastic processes to processes with nonseparable range.

Theorem 4.3 Let L C *18 be internal. Then I : n x S ---+ t is Pettis measurable, il and only il I has a weak lifting F : n X * S --+ L.

Proof: Because the "if" -part of the theorem is well known, we only prove the "only if"-part. By Lemma 4.1, for each cp E 18' there exists an A-measurable function G", : n x * S ---+ ·ID such that for all a E IN x IN and J1-a. a. wEn (J1-a. a. depends on cp):

rp 0 I(w) E Wa ~ cp 0 F(w) E *Wa.

For all n E IN and cp E 18' there exists X n ,,,, E A such that p.(n) - lin < p.(Xn ,,,,)

and

X n ,,,, C {w E n I Va E IN x IN(rp o l(w) E Wa

~ G",(w) E *Wa and rp 0 I(w) is continuous )}.


Notice that for all wE X n,,,, rpo/(w,Os) RJ G",(w,s) for all nearstandard s E ·S. Our aim is to find an A-measurable function F : n --t {G : "S --t LIG internal} such that for all cp E 18' and all n, m, k E IN and all open G :::> Um

(+) F[G;l["W(n,m)] n X k ,,,,] C {H:"S --t LIH internal and cpo HE ·W(On,G)}.

By saturation, it suffices to show that for all finite 4? C 18' and E C IN

n{G;l['W(n,m)] nXk,,,,ln,m,k E E and cp E 4?}"I- 0 ~ n{{H :"S --t LIH internal and cp 0 HE "W(On, G)}I

n, m E E, cp E 4? and G :::> Um is open} "I- 0.

Let wEn be an element of the left hand side. Then G",(w) E 'W(n,m) and wE Xk,,,, for all cp E 4? and all n, m, k E E. It follows that rp 0 I(w) E W(n,m) for all cp E 4? and all n, m, k E E. By Lemma 4.2, there exists H : • S --t L such that for all cp E 4? and all n, m E E and all open G:::> Um

cp 0 HE 'W(On, G).

This proves that the right hand side is nonempty and therefore a function F with the property (+) exists. It is easy to check that rp 0 I (w, ° s) RJ cp 0 F( w, s) for all w E U X n ,'" and all nearstandard s E "S .•

5 Bochner integrable functions

Recall that I : n --t 10 is called Bochner integrable, if there exists a sequence (fn)nEIN of L(A)-measurable simple functions In : n --t 10 converging to f in measure and such that

lim In IIln - ImlldtJ = O.

Then the Bochner integral of f over B E L(A) is defined by

B -l IdtJ = lim l IndtJ·

The internal counterparts to the Bochner integrable functions are the internal SBochner integrable functions on n with values in "10. An internal A-measurable function F : n --t "10 (recall that F is then *finitely valued) is called S-Bochnerintegrable, if IIFII is S-integrable, i. e., for all k E "IN \ IN

r 1IFlldll RJ 0, where Ak := {IIFII ~ k}. lA. Recall that IIFII is S-integrable, if and only if

VA EN n A(l IIFlldll ~ 0) and lllFlldll is limited.

In [12] there is a proof of the next result:

82 H. Osswald

Proposition 5.1 A function f : n --+ ID i s Bochner integrable, if and only if f has an S-Bochner integrable lifting F : n --+ *ID {i. e., {t ¢ F} is a p,-nullset), in which case

8 -l fdp, ~ i Fdp, for all BE L(A) and allp,- approximations A of B.

It is easy to see that we get as in the scalar valued case:

If F : n --+ *ID is A-measurable and a lifting of a Bochner integrable function f, then there exists H E *IN \ IN such that

FI : w ~ {F(W) ,if IIF(,,:)II $ H for all wEn H 0, otherwIse

is S-Bochner integrable. Of course, FIH remains a lifting of f.

6 Dunford and Pettis integrable functions

Now we will prove lifting theorems for Dunford and Pettis integrable functions.

A function I : n --+ 113 is called Dunford integrable, if cp 0 I is Bochner integrable for all ({J E 18'. An A-measurable function F : n --+ *18 is called S-Dunford integrable, if ({J 0 F is S-Bochner integrable for all ({J E 18'.

The proof of the next result - it is a lifting theorem for Dunford integrable functions - is a simplification, due to P. Loeb, of a proof for the same proposition in an earlier version of this paper.

Proposition 6.1 Let L be an internal subset of *18 and I : n --+ t. Then I is Dunford integrable, if and only if f has an S -Dunford integrable weak lifting F: n --+ L.

Proof: The "if" part follows from 5.1. Now we prove the "only if" part. By Theorem 4.3, I has a weak lifting G : n --+ L. For every ({J E 18' there exists an Hcp E *IN \ IN such that (({J 0 G)IH", is S-Bochner integrable. Hence, for every ({J E 18' there exists a p,-nullset Ncp E A such that

lO\N", . ({J 0 G = (({J 0 G)IH",.

By saturation, there exists an N E AnN with U{Ncpl({J E 18'} c N. Since ({J 0

(lo\N . G) = lO\N . ({J 0 G, the function F := lO\N . G satisfies the assertion .•

The next result is a lifting theorem for Pettis integrable functions. A Dunford integrable function I : n --+ 18 is called Pettis integrable, if for all B E L(A) . , there exists a vector aB E 18 such that for all ({J E 18

cp(aB) = 8 -l cp 0 Idp,.


The vector aB is called the Pettis integral of lover B, written

An S-Dunford integrable function F : n ---+ *IB is called S-Pettis integrable, if fA Fdp, is nearstandard for all A EA.

Theorem 6.2 Let L C *IB be internal and I : n ---+ L. Then I is Pettis integrable, il and only il there exists an S-Pettis integrable weak lilting F : n ---+ L 01 I, in which case

P -Is Idft ~w 1 Fdp, for all BE L(A) and all p, - approximations A 01 B.

Proof: Let I be Pettis integrable. Then I is Dunford integrable, hence, by 6.1, I has an S-Dunford integrable weak lifting F : n ---+ L. We obtain for all 'I' E IB', for all B E L(A) and all p,-approximations A of B

cp(P -Is Idp,) = B -Is cp 0 Idft ~ 1'1' 0 Fdp, = '1'(1 Fdp,).

Hence fA Fdp, is nearstandard and P - fA Idft ~w fA Fdp,.

Now assume that F is an S-Pettis integrable weak lifting of I. By 6.1 I is Dunford integrable. Let BE L(A). Then there exists a p,-approximation A of B. Define

aB := 1 Fdp, E lB.

We obtain for all 'I' E IB'

Hence, I is Pettis integrable .•

Remark: Theorem 6.2 reminds of a result in the standard theory of Pettis integrability, due to R. F. Geitz [5]. He proved that a bounded function I : n ---+ IB, where IB is a Banach space and n is a perfect finite measure space, is Pettis integrable, if and only if there exists a uniformly bounded sequence (fn)nelN of measurable simple functions In such that for all 'I' E IB'

lim 'I' 0 In = 'I' 0 1ft-a. e. (ft - a. e. depends on '1') In Theorem 6.2, F can be understood as an idealization of Geitz's sequence, which exists for all Pettis integrable functions on Loeb spaces. Notice that Loeb spaces belong to the class of perfect spaces.

84 H. Osswald

7 Examples

In this section, we will give 6 examples to show, how the preceding results can be used.

(a) The set of values of the indefinite Bochner integral of a Bochner integrable function f: n -+ ID (i. e., {B - IE fdfJ,IB E L(A)}) is norm compact.

(f3) Suppose that f : n -+ 18 is Pettis integrable. Then the set of values of the indefinite Pettis integral of f is weakly compact.

Both results are well known. See [4]. Here they follow immediately from Proposition 5.1 and Theorem 6.2 respectively, and from W. A. J. Luxemburgs result, quoted in 3.1(-y).

(,) Convergence theorem for uniformly Pettis integrable functions

In this example we combine the results in the preceding sections with nonstandard techniques, applicable to uniformly integrable functions. See for example N. Cutland [3].

A sequence (fn)nelN of Pettis integrable functions fn : n -+ 113 is called uniformly Pettis integrable, if for all cp E IB'

lim r Ilr,O 0 fnlldfJ, = 0 uniformly in n E IN. k-+oo J{II";Ofnll~k}

A sequence (Fn)nelN of S-Pettis integrable functions Fn : n -+ ·IB is called uni· form.ly S.Pettis integrable, if for all cp E 18'

lim 0 r IIcp 0 FnlldfJ, = 0 uniformly in n E IN. k-+oo J{lIcpoFnll~k}

Lemma 7.1 A sequence (Fn)nelN of S-Pettis integrable functions Fn : n -+ ·IB is uniformly S-Pettis integrable, if and only if (Fn)nelN can be extended to an internal sequence (Fn)nE*1N such that Fn is S-Dunford integrable for all n E ·IN

Proof: First we will prove the "only if" part. By the hypothesis, there exists a standard function g : IB' x IN -+ IN such that for all n E IN and all cp E IB'

By saturation, (Fn)nelN can be extended to an internal sequence (Fn)nE*1N such that all Fn are A-measurable and for all n E IN and all cp E IB'


Hence, if k E *IN\IN, then k ~ *g( cp, n) for all n E IN and cp E IS', thus all FI , l E "IN, are S-Dunford integrable.

To prove the "if part", assume that the assertion is wrong. Then there exist cp E IS', a standard € > 0 and arbitrary large k E IN such that

By saturation there exists K E *IN \ IN such that

which contradicts the hypothesis that cp 0 Fn is S-Bochner integrable for all n E "IN .

• Let (fn)neIN be a sequence of Pettis measurable functions fn : 0 --+ lB. We say

that (f n) converges to f : 0 --+ 113 in measure, if (rp 0 f n) converges to cp 0 f in measure. Note that f is then Pettis measurable.

Using Theorem 6.2 and Lemma 7.1 we obtain immediately

Proposition 7.2 Suppose that 113 is weakly complete. If (In) is a sequence of uniformly Pettis integrable functions fn : 0 --+ 113, converging to f : 0 --+ 113 in measure, then f is Pettis integrable and for all BE L(A)

p -l fd{l = lim P -l fnd{l in the weak topology.

In the remainder of this paper we make the following assumptions:

L is an internal "convex subset of *18 ,'containing only nearstandard points and the origin o. Let (0, A, J.£) be an internal probability space, where 0 is a ·finite set. Because then A is also "finite, we obtain for all w E 0

[w] := n{A E Alw E A} E A..

[w] is called the atom of w. Then the measure J.£ results from an A-measurable weighted counting measure clJ ' given by

where I . I denotes the internal "finite cardinality of .. It is easy to see -write A as the disjoint union of all atoms of A- that

J.£(A) = L clJ(w) for all A E A.. weA

86 H. Osswald

Let us identify IL and Cw Then

IL(A) = E IL(W) and 1 FdlL = E F(w) 'IL(W) for all A E A, weA A weA

where F is an A-measurable function on 0 with value in '18 or ·ID.

(8) Keisler's Fubini Theorem

Assume that A = 'P(O) (so L(A) = L(O)), and suppose that (A, L(A), v) is a second Loeb space over an internal probability space (A, 'P(A), v) with A *finite. Define

lL'v:= E IL(W)' v(>.) for all internal subsets A cOx A. (w,>.)eA

R. A. Anderson [2] has shown that the Loeb product (0 x A, L(O x A), IL 7>.) of o and A is an extension of the completion (0 x A, L(O x A), P, ® v) of the product of 0 and A in the standard sense. Because this extension may be strict -examples were given by D. Norman and D. H. Hoover- standard Fubini Theorem cannot be applied to the Loeb product. So, H. J. Keisler [9] proved a Fubini Theorem for the Loeb product. The next Proposition is a vector valued version of Keisler's result.

Proposition 7.3 Suppose that f : 0 x A --+ L is Pettis integrable w. r. to p. Then there exists a Pettis integrable function h : 0 --+ L such that for all cp E 18'

tP 0 h(·) = B - [tP 0 f(·, >.)dv(>.) p,-a. e. (p,-a. e. depends on cp)

and P - f hdp, = P - f JdP. io ioxA

Proof: By Theorem 6.2, there exists an S-Pettis integrable weak lifting F ; 0 x A --+ L of f. Since for all W E 0

H(w) ;= E F(w, >.) . v(>.) >'eA

is a 'convex combination of elements F(w,>.) E L,H(w) E L. Hence, h(w) ;=

H(w) E L is well defined. Fix cp E 18'. Using Keisler's Fubini Theorem, it is easy to see that for p,-almost all w E 0, cp 0 F(w,·) is an S-Bochner integrable lifting of tP 0 f(w, .). Hence, by Proposition 5.1, for p,-almost all wE 0

tP 0 h(w) = 0cp 0 H(w) = B - [tP 0 f(w, >.)dv(>.).

It remains to prove that h is Pettis integrable; Fix cp E 18'. Since cpoF is S-Bochner integrable

" IIcp oH(w) 1I'IL(w) < " IIcpoF(w, >.) 1I'IL(w) .v(>.) { ~ li~' i~tAdi~ a IL-nullset L...J - L...J IS ml e III any case. weA (w,>.)eAxA


Hence, cp 0 H is S-Bochner integrable and of course a lifting of rp 0 h. Since

f HdJ1- = EH(w). J1-(w) E L, f HdJ1- is nearstandard. iA weA iA

By Theorem 6.2, h is Pettis integrable and we obtain

p - f f#><v ~w E F(w, >.). J1-(w), v(>.) ~w P - f hdjJ, .• inxA (w,'\)enxA in

(f) Existence of conditional expectation

Stochastic filtrations, which result from internal ones, play an important role in the nonstandard approach to stochastic analysis. Se e for example [1] or [9]. We assume now that J1-(A) 1= 0 for all A E A.

Proposition 7.4 Let f : n ---t t be Pettis integrable and let B be an internal subalgebra of A. Then there exists a Pettis integrable function 9 : n ---t t w. r. to (L(B),jJ,) such that

p -fa fdjJ, = P -fa gdjJ, for all BE L(B).

Proof: By Theorem 6.2, f has an S-Pettis integrable weak lifting F : n ---t L. Recall the definition of the atom [w] of wand define

G(w):= r FdJ1-' J1-([w])-l for all wEn. i[w]

Since L is ·convex, G(w) E L for all wEn. It is easy to see that for all A E B

E G(w)· J1-(w) = E F(w)· J1-(w). weA weA

Moreover, Gis B-measurable. It follows that Gis S-Dunford integrable and a weak lifting of 9 : n ---t t, where g is defined by g(w) := G(w). Hence, by Proposition 6.1, 9 is Dunford integrable. Since EweA G(w) . J1-(w) is nearstandard for all A E B, by Theorem 6.2, 9 is Pettis integrable w; r. to (L(B),jJ,) and we obtain for all B E L(B) and all J1--approximations A E B of B

P- r fdjJ,~wEF(w)'J1-(w)=EG(w)'J1-(w)~wp-lgdjJ, .• iB weA weA B

In the standard theory of Pettis integrability,' W. R. Pestman [14] proved the existence of conditional expectation for Pettis integrable functions with values in metrizable convex compact sets and extended this result to totally summable functions.

88 H. Osswald

A proof of the existence of conditional expectations with values in measure convex sets can be found in the lecture notes of G. Winkler [18].

(() Nondeterministic Peano Caratheodory integral equations

In this application, we use the well known close relationship between Loeb and Lebesgue measure. Let H be an infinite natural number. Define

D.t := H-1, T:= {O, D.t,2· D.t, ... , I},

II(A) := IAI . D.t for all internal AcT.

(T, L(T), v) denotes the Loeb space over (T, 'P(T), II).

The next result is due to H. J. Keisler [9]. Here .x denotes Lebesgue-measure and Leb([O,l]) the Lebesgue-measurable subsets of [0,1]).

Theorem 7.5 A function 9 : 0 X [0,1] -+ ID is L(O)i8lLeb([O, l])-measurable, if and only if 9 : (w, t) f---t g(w, °t) is L(O x T)-measurable.

A function I : 0 X [0,1] -+ 18 is called a Pettis integrable process, if for all <p E 18' and it-almost all w E 0 (it-a. a. depends again on <p)

cp ° I (w, .) is Bochner integrable

and if there exists a function x : 0 X [0,1] -+ 113 such that for all <p E 18'

(-)

cp 0 x(w,.) = / cp 0 I(w, s)d.x(s) for it - a lmost all wE O.

o

The mapping x = x(o,.) is then called the Pettis integral process of I, written

(-)

x = P - / f(o, s)d.x(s).

° The Pettis integral process of f is only unique w. r. to weak equivalence. Two functions x, y : 0 X [0,1) -+ 18 are called weakly equivalent, if for all <p E 18'

cp 0 x(w,·) = cp 0 y(w,·) for it - almost all wE O.

For the next proposition, fix 'IjJ E 18' and assume that there exists a closed subset S of ID such that

'IjJ[L) C • S, so ~[l) c S.

Because we are able to write the Pettis integral as a sum, we may apply a method, developed by H. J. Keisler, and used by many authors in order to solve differential


equations -see for example [9J-, to handle integral equations of the form below. The method is working as follows:

First represent the differential equation by a 'finite difference equation. In this step, it is crucial to characterize standard measurable and integrable functions by objects which behave like finite entities. Here we use the preceding lifting theorems.

The second step may be the simple part of that method: Solve the difference equation.

In the third step show that this solution is -up to an infinitesimal error- a standard solution of the equation, we have started from.

By means of the preceding lifting results this method may be applied to the following

Proposition 7.6 Suppose that

and that

f : n x [0, 1 J x S ~ t and g: n x [0, IJ ~ t

(a) f is Pettis measurable w. r. to (L(n)®Leb[O, IJ, [L ® A) and

(b) 9 is Dunford integrable w. r. to (L(n)®Leb[O, IJ,[L ® A) and

(c) for all 'P E 18' and [L-almost all wEn ([L-a. a. depends on 'P)

sup IIcp 0 f(w,·, s)11 ~ IIcp 0 g(w, ')11· sES

The assertion is: There exists a Pettis measurable function (w. r. to (L(n),[L)) x: n x [O,IJ ~ t such that

x(w, t) = P -lot f(w, S, ~ 0 x(w, s))dA(s),

i. e., x is the Pettis integral process of f(o,.,~ox(o,.)).

References

[IJ S. Albeverio, J. E. Fenstad, R. H91egh Krohn, T. L. Lindstr91m, Nonstandard methods in stochastic analysis and mathematical physics; Academic Press (1986)

[2J R. A. Anderson, A nonstandard representation of Brownian motion and Itointegration; Israel J. Math. 15 (1976)

[3] N. J. Cutland, Nonstandard measure theory and its application; Bull. London Math. Soc. 15 (1983)

90 H. Osswald

[4] J. Diestel, J. Uhl, Progress in vector measures 1977 - 83; Lecture notes in Math. 1033 (1983)

[5] R. F. Geitz, Pettis integration; Proc. Amer. Math. Soc. 82 (1983)

[6] S. Heinrich, Ultrapowers of locally convex spaces; Mathematische Nachrichten 118 (1984)

[7] C. W. Henson, L. C. Moore, The nonstandard theory of topological vector spaces; Trans. Amer. Math. Soc. 172 (1972)

[8] A. E. Hurd, P. A. Loeb, An introduction to nonstandard real analysis; Academic Press (1985)

[9] H. J. Keisler, An infinitesimal approach to stochastic analysis; Memoirs Amer. Math Soc. (1984)

[10] P. A. Loeb, Conversion from standard to nonstandard measure spaces and applications to probability theory; Trans. Amer. Mat. Soc. 211 (1975)

[11] P. A. Loeb, A functional approach to nonstandard measure theory; Contemporary Math. 26 (1984)

[12] P. A. Loeb, H. Osswald, Nonstandard integration theory on topological vector spaces; to appear

[13] W. A. J. Luxemburg, A general theory of monads; Applications of model theory to algebra, analysis and probability, Holt Rinehart and Winston, New York (1969)

[14] W. R. Pestman, Convergence of martingales with values in locally convex Suslin spaces; Math. Institut, Rijsuniversiteit Groningen, preprint (1981)

[15] B. J. Pettis, On integration on vector spaces; Trans. Amer. Math. Soc. 44 (1936)

[16] D. A. Ross, Lifting theorems in nonstandard measure theory; Proc. Amer. Math. Soc. 109 (1990)

[17] M. Talagrand, Pettis integrability; Memoirs Amer. Math. Soc (1984)

[18] G. Winkler, Choquet order and simplices; Springer Lecture notes in Math. 1145 (1986) .

A COUNTEREXAMPLE TO THE SPECTRAL MAPPING THEOREM REVISITED FROM A NONSTANDARD POINT

OF VIEW

H. Ploss Zentrum fiir Fernstudien der J. K. UniversiHit Linz

Belruptstr. 10 A - 6900 Bregenz

Abstract

We show that a classical counterexample to the spectral mapping theorem for the exponential function is not very surprising if we consider the example in a suitable nonstandard setting.

Keywords: Spectral mapping theorem, Co-semigroups

1 The Counterexample

We recall briefly the example of Zabcyk and refer to [5], [1] and [3]. [3] is also a general reference for one-parameter semigroups of operators.

For every n E IN consider the n-dimensional Hilbert space a::n and operators An E L(a::n ) defined by the matrices

o 1 o o 0 1 0

o o

Then a(An) = {o}, IIAnl1 = 1, Ann = O.

Consider now the Hilbert space E

A:= E9 (An + 27rin) with maximal domain in E. nEIN

Analogously we define T = (T(t))t~O by

1 o

E9 a::n and the operator nEIN

T(t):= E9 (e2".intexp(tAn)). nEIN

91

92 H. Ploss

Since II exp(t An)ll ::; et for every n E IN, t ~ 0 and since t ~ T(t) x is continuous on each component a;n it follows that T is a strongly continuous semigroup of operators.

Its generator is the operator A as defined above.

The spectrum of A and the resolvent of A are given by

a(A) = {27l'in: n E IN} and

n(,\, A) = ~ n(A, An + 27l'in), ,\ E e(A) = a; \ a(A) nEIN

respectively.

Therefore for the spectral bound s(A) := sup{ReA : A E a(A}} of the generator A we have s(A) = O. On the other hand one can compute that the growth bound

w(T):= inf{wE IR : IIT(t)lI::; Mwewt , for aUt ~ o and suitableMw }

of the semigroup T is in the example given by w(T) = 1.

From this it follows (e.g.[3]) that the spectral mapping theorem (SMT)

a(T(t)) \ {O} = exp (t.a(A)) = {eAt: ,\ E a(A}}

cannot be true in the example of Zabcyk. We notice that in [1J the spectra of both the generator and the semigroup are computed explicitly by means of

Gearhart's Theorem: Let T = (T(t))t>o be a semi group with generator A in a

Hilbert space. Then eA E e(T(1)) iff -

i) ,\ + 27l'in E e(A) for all n E 71., and

ii) sup {lIn(A + 27l'in, A)II : n E 71.} < 00

In particular: {e": IAI::; 1} = a(T(1)) is shown in [1J.

2 The example from a nonstandard point of view

As a general reference for our nonstandard terminology we refer to [L]. For n E ·IN denotes 12(n) the space of all internal sequences Xl!'" ,Xn of hypercomplex numbers given the metric

n 1

d(x,y) = Ilx - ylb = (E Ix; - y;1 2r ;=1

We fix w E ·IN"" and define

w

X := ~ 12(n) n=l

A counter example to the spectral mapping theorem 93

For every n, 1 ~ n ~ w, consider the internal linear (over Star<V) operators An : 12(n) -+ 12(n) defined by matrices as in part 1. and define

w

A := E9 (An + 21rin). n=l

( The internal operator A : X -+ X can be regarded as an 0 x O-matrix. )

For A E Star<V \ {21rin: 1 ~ n ~ w} there exists

R(A, A) .- (A - Atl given by w

R(\A) = E9R(A,An +21rin) with n=l

n-l

R(\An + 21rin) = R(A - 21rin,An) = ~)A - 21rintk-1Ank, 1 ~ n ~ w.

k=O

By means of an easy estimation we see that for A with st(ReA} > 1, R(A, A} is bounded.

But we claim that for n E "]Noo, n ~ w and A satisfying IA - 21rinl ~ 1 (A ¥- 21rin) R(A, A} is unbounded.

To verify this consider en = (0, ... , If E 12(n) or en embedded in X respectively. Then with 11 := A - 21rin we have

n 1

/IR(\ A} en/l = /IR(Il, An} en/l = (L 11l1-2k ) 2. k=l

Therefore it is natural to say that A = 21rin + 11, for n E "]Noo, n ~ w, 1111 ~ 1 is a spectral value of A and consequently the spectral bound of A is equal to 1.

For t E "ffi.+ let

n-l A k tk Tn(t} . - exp(21rin t} L +-' 1 ~ n ~ w .

k=O

n-l /IAn/lk tk W tk et . Then /ITn(t}/1 < L k! ~ Lk! ~

k=O k=O

W

Consider T(t} := E9 Tn(t}. n=l

Because of /IT(t}/i = max{/ITn(t}/i : 1 ~ n ~ w}, T(t} : X -+ X is for finite t a bounded operator. Clearly (T(t})tE'tarIR+ inherits the semigroup property

T(O) T(t + s)

Idx ,

T(t}T(s).

94 H. Ploss

For simplicity we choose t = 1. Then by direct computation [ or using the formula t

(e~t - T(t)) I = (>. - A) f e~(t-B)T(s) Ids, I in the underlying Banach space, and o

a transfer argument] we conclude that with >. = 211'in + IL

IIT(I) 1<.(>', A) e .. - ejJ 1<.(>., A) enll is finite, 1 ~ n ~ w.

'I1'::nh-'-.....t-""T • Then IIlnll = 1 and for n E *INeo, n ~ w, IILI ~ 1

IIT(I) I .. - ejJ 1 .. 11 ~ o.

Therefore given e E 14, there exists N E IN such that

In order to get a standard setting let S be the set of all Xl. X2, ••• , xm , 0, ... ,0 where both m and all xi's are finite and take the nonstandard hull

(pns(X, S)/fl$' d)

which is isomorphic to 12 (see e.g. [L, III,3)).

Because of T(t): pns(X, S) --+ pns(X, S), for finite t E *R+, we can define T(t): 12 --+ l2 in the usual way. Moreover it is easy to see that

T(t) x ~ x, for x E pns(X, S) and t positive infinitesimal.

From this it follows that (T(t))t>o is a strongly continuous semigroup on l2' see [41· -Clearly this semigroup coincides with the semigroup in the example of Zabcyk. Using (*) we conclude that for IILI ~ 1

(approximate point spectrum).

That means we have verified the result in [11, but we did not need Gearhart's Theorem.

3 Final Remark

We have shown that from a nonstandard point of view the example of Zabcyk is not really a counterexample to the SMT. We have only to approximate the standard setting from above. In this way we find the points in the spectrum 01 the generator in the extended complex plane, which - via the exponential function - contribute to the spectrum of the semigroup. We are working to transfer these arguments into a more general situation.

A counter example to the spectral mapping theorem 95

References

[1J Hejtmanek, J., Kaper, H.: 1986, "Counterexample to the Spectral Mapping Theorem for the Exponential Function", Proc. Amer. Math. Soc. Vol. no. 96, pp. 563-568.

[2J LindstrliSm, T.: 1988, "An Invitation to Nonstandard Analysis", in Cutland, N. (ed): "Nonstandard Analysis and its Applications", London Math. Soc. Stud. Texts Vol. no. 10, pp. 1-105

[3J Nagel, R.: 1985, "One-Parameter Semigroups of Positive Operators", Leet. Notes Math. Vol. no. 1184, Springer, Berlin-Heidelberg- New York

[4J Wolff, M.: 1984, "Spectral Theory of Group Representations and their Nonstandard Hulls", Israel. J. Math. Vol. no. 48, pp. 205-224.

[5J Zabcyk, J.: 1975, "Note on Co-Semigroups", Bull. Acad. Polan. Sci. Vol. no. 23, pp. 895-898.

Nonstandard polynomials in several variables

Hermann Render Department of Mathematics, University of Duisburg, Lotharstr. 65

D-47057 Duisburg, Germany

Abstract

c. Impens has investigated nonstandard polynomials in one variable. In this note we show that many of his results carryover to polynomials in several variables. The concept of complex and real micro continuity of nonstandard polynomials is intimately related to polynomial approximation in function theory.

1 Introduction

In his book "Non-standard Analysis" A. Robinson has given a very nice discussion of • -holomorphic functions of one complex variable. Of fundamental importance is the concept of S-continuity or complex microcontinuity: If U is an open subset of cr:n then an internal function I: ·U ~ .cr: is called S-continuous or complex microcontinuous at wE U if z ~ w implies I(z) ~ I(w). Let fin·cr: be the set of all finite numbers of .cr:. Then the set

Sf:= {w E U: I(w) E fin·<r and 1 is complex microcontinuous at w} (1)

is called the set 01 standard complex microcontinuity. It is well-known that the standard part st 1 : Sf -+ cr: of the internal function I, defined by st 1 (w) : = st (f (w)), is a continuous function; in the last equality st is the standard part map of the Hausdorff space cr:. We denote the set of all holomorphic functions on an open subset U of cr:n by H(U), see [7] for the precise definition. An element 1 E • H(U) is called a ·-holomorphic function. In analogy to results due to A. Robinson and C. Impens in the one-dimensional case we prove that a function J E • H(U) is complex microcontinuous at w E U satisfying J(w) E fin·cr: iff J(z) E fin·cr: for all z ~ w. If Q: .cr:n -+ ocr: is a nonstandard polynomial (for definition see section 2) then Q(z) = l:"EE a,,(z - w)" is complex microcontinuous at w E cr:n iff la"II/IIII E fin·cr: for all /J E E. At a first glance one may be tempted to consider the set of standard complex microcontinuity as the nonstandard counterpart of the domain of convergence Cp of a power series P( z), i.e. the interior of the set of all z E cr:n such

96

Nonstandard polynomials in several variables 97

that the power series P(z) converges. The following simple example shows that this is not the case: Let p(z) = z(z - 3) and Q(z) = p(z)N for some infinitely large N E ·IN. Using Theorem 2.3 (ii) =} (i) it is easy to see that SQ consists of all z E <en with Istp(z)1 < 1. Then 0 and 3 are contained in SQ which is also a subset of B1(0) U Bl(3), where Br(w) .- {z E <e : Iz - wi < r}. In particular SQ is not connected.

The main difference between nonstandard polynomials and power series can be expressed in the following way: complex microcontinuity at a point represents like a power series local holomorphy but complex micro continuity on the set SQ represents a special kind of polynomial approximation of the standard part st Q.

The paper is divided in four sections. In the second section we generalize results about complex microcontinuity to the case of several variables. In the next section we consider hyperfinite truncations of power series (in one variable): the domain of convergence is the intersection over all hyperfinite truncations of their standard sets of complex microcontinuity. Moreover we give a standard description of the union of all SQ where Q runs over all truncations of a power series. The last section is devoted to the question of the structure of SQ for a nonstandard polynomial in one variable. We give an easier proof of a result of C. Impens that every simply connected open set U c <e is the set of standard complex micro continuity for an appropriate nonstandard polynomial. Using the theorem of Mergelyan we construct a plenty of certain nonstandard polynomials which are micro continuous on a subset • K where K is a rather general compact subset of <e ..

Finally we mention some terminology: 1R+ denotes the set of all positive real numbers. By [a, b] we mean the set of all real numbers x with a :$ x :$ b. The set of all infinitely large hypernatural numbers is denoted by ·lNco • We say that z E .<en is infinitesimal near to wE <en, briefly z ~ x, if Iz-wl < E for all E E IR+. The set ns·U of all nearstandard points is the union of all monads m( w) := {z E • U : z ~ w} with w E U.

2 Complex micro continuity of nonstandard polynomials

Let IN be the natural numbers including zero and let n E IN be a fixed natural number greater equal than 1. A multi-index v = (VI>'" , vn) is just an element in INn and as usual we define Ivi := VI + ... + vn . It is convenient to embed IN into INn by the formula k r-+ (k, ... , k). We can introduce a partial order on INn if we define IJ :$ V to mean that lJi :$ Vi for i = 1, ... , n. An element V E .lNn is called infinitely large if ·IJ ·:$v for alllJ E INn. The set of all infinitely large multi-indices is equal to ·lNco x ... ·lNco , or more briefly, to ·IN~. Let z = (Zl>"" zn) E <en, w = (Wl>"" wn) E <en and V = (VI>"" vn) a multi-index. If av E <e and av #- 0 then av(z - w)V := av(zl - wdllJ. . '" . (Zn - wn)Vn is called a monomial. A polynomial

98 H. Render

(in n variables) is a function Q: a::n -+ a:: of the form Q(z) = LVEE avzv where E is a finite subset of INv and av E a::. The set of all polynomials in n variables is denoted by P(a::n). Every element in *P(a::n) is called a nonstandard polynomial. It follows by the transfer principle that every nonstandard polynomial can be written in the form Q( z) = LVEF bv( z - w)V, where W E *a::n is given and F is a suitable hyperfinite subset of *lNn and bv E *a::.

Let W = (Wb ... , wn) E a::n and (rl,"" rn) E R~. A polydisc Pr(w) is a set of the form Brl (Wl) x ... x BrJ wn) where Br, (Wi) are the open balls {a E a:: : la - Wi I < r;} in the complex plane for i = 1, ... n. The topological closure of Pr(w) is called the closed polydisc and is denoted by Pr(W). Note that Pr(w) = {z E a::n : IZi - wil :::; ri for i = 1, ... , n}. For r E R+ we write Pr(w) instead of p(r, ... ,r) (w).

2.1 Definition. Let Q(z) = LvI'S' av(z - w)V be a nonstandard polynomial. We define CQ := CQ(w) = sup{stlavi/ivi : vEE with Ivl E *lNoo } and we call HQ := 1/CQ the Hadamard number (at the point w). In the case of CQ = 0 we define HQ = 00.

In the last definition it is assumed that st is the standard part map with respect to the one-point compactification of the Hausdorff space a::. Hence, if some number lavl 1/ v (with Ivl E *lNoo ) is not contained in fin * a:: then HQ is equal to O. The next result shows that nonstandard polynomials behave in a certain way like power series.

2.2 Theorem: Let Q(z) = LV>EE av(z - w)V be a nonstandard polynomial. If a'/J E fin *a:: for every J1, E EnlNn then for every 0 < r < HQ and for all z E * Pr(w) the number Q(z) is finite and

stQ(z) = L stav·st(z-wt· vEEnlNn

(2)

Proof: For HQ = 0 the statement is trivial. If HQ > 0 then CQ < 00. Let r < HQ and choose (J E R with CQ < (J < 1/r. Let *lNk := {I E *IN : l ~ k} for k E IN. Then nf:l(*lNk x *IN x ... x *IN) n E is contained in the internal set {v E E : lavll/lvl :::; (J}. By a saturation argument there exists k E IN such that lavll/lvl :::; (J for all v = (Vb'" ,vn ) E E with k :::; Vl' Using this argument for every component we obtain a finite set F C EnlNn such that lavll/lvl :::; (J for all v E E\ * F. We can assume that E\F is contained in the set {v E E: k :::; Vi for all i = 1, ... , n} for a given large number k E IN. Moreover there exists (j E *lNn with v :::; (j for all vEE. We are looking at the inequality

(3) vEF vEE\F

for z = (Zb ... ,zn) E *a::n satisfying IZi - wil :::; r for i = 1, ... ,n and we obtain as


an upper bound

(4)

Since ~v<IT(fJ . r)lvl ::; I17=1(~~:=o(8. r)Vi) = I1:':1 (8r~:i~:-1 E fin *<e it follows that Q(z) is finite. Eq. 2 follows now from the equations 3 and 4 since we can choose k E IN arbitrary large .•

Let U be an open subset of <en and f E H(U). Let Ii: la, bl --+ <e be continuous paths for i = 1, ... , n and set I := (/1. ... , In): la, bln --+ <en. We define the iterated integral over I by the formula

If r E R~ and W = (Wl,""Wn ) E <en we define Cri,w,(t) := Wi + rieit and cr,w := (cr1,Wl"'" cr.,w.): [0, 27Tln --+ <en. Moreover we denote the function z = (Z1." . , zn) ~ (Zl - wd· ... · (zn - wn) by (z - w)l; hence we consider 1 as the multiindex (1, ... , 1). Using the Cauchy integral formula in the one-dimensional case it is not very hard to prove the well-known Cauchy integral formula for f E H(U): If Pr(W) is a subset of the open set U c <en then for all Z E Pr(w)

(6)

C. Impens has proved the equivalence of the following statements for nonstandard polynomials in one variable:

2.3 Theorem: Let Q(z) = ~vEEav(z - w)V be a nonstandard polynomial and W E o::n. Then the following statements are equivalent:

(i) There exists r E R+ such that Q(z) E fin *<e and Q is complex microcontinuous for all z E Pr(w).

(ii) Q is complex microcontinuous at wand Q( w) E fin *0::.

(iii) Q(z) E fin *<e for all z ~ w.

(iv) The nonstandard monomial av(z - wy is complex microcontinuous at w for every vEE.

(v) lavll/lvl E fin *<e for aU v E E.

(vi) Q(z) := ~vEE lavl(z - w)V is complex microcontinuous at w.

Proof: The equivalence of (i), (ii) and (iii) is valid for every element f E * H(U): we have only to show (iii) => (i). The underflow principle applied to the set {k E *IN :

100 H. Render

(Vu E *P!.(w)) (If(u)l:5 k)} yields k E IN such that If(u)l:5 k for all u E *P!.(w). • •

Now let r = (1/2k, ... , 1/2k) and u E *Pr(w) and v E Pr(w) with u i:::! v. Then the Cauchy integral formula shows that for p := 2r

* 11 (e-*v)l-(e-u)l If(u) - f( v)1 :5 (21T)nl Cp, .. f(e)· (e _ u)1(e _ *v)1 del. (7)

Since lei - uil ~ 1/2k and lei - *vil ~ 1/2k and If(e)1 $ k on *Pp(w) we obtain

If(u) - f( *v)1 :5 (2!)n . (2:)2n . k '1., .. I(e - *v)l - (e - u)llde. (8)

By Lemma 2.4 we know that I(e - *v)l - (e - u)ll :5 € for all € E 1R+ Hence If(u) - f( *v)1 i:::! 0, i.e., that 1 is complex microcontinuous at v E Pr(w).

Now let us prove (i) ~ (iv). Let Cr,w be as above. By the transfer principle the meaning of cr,w for r E *1R.+ is clear. Then we can define a map < ',' >: * p(Vn) x * p(Vn) --+ *(V by defining:

r21r r21r < I,g >:= 10 ... 10 I(c,.,w(t)). g(cr,w(t))dt,

where t = (tl , ... , tn) E *[0, 21T]n. Moreover we have that

for v =I- p, for v = p,.

(9)

(10)

Define hv(z) := (z-w)Vav/ (rlvl(21T)nlavl) for v with av =I- 0, and hv(z) = 0 otherwise. Then Eq. 10 means that the monomials hI' are orthogonal and we obtain for v =I(0, ... ,0)

(11)

Obviously we have Ihv(cr,w(t))1 :5 1, and for ri:::!O it follows by complex microcontinuityat w that Q(cr,w(t)) - ao i:::! O. Eq. 11 yields lavlr1v1 i:::! 0 for all r i:::! O. Since lav(z - w)vl :5 lavlr1v1 with r := max{lzI - wII, ... , IZn - wnl} we obtain (iv). The implication (iv) ~ (v) is very easy: if M:= lavll/lvl is infinitely large then r:= l/M is infinitesimal near to zero. But lavl·r1vl = 1, a contradiction to (iv). For (v) ~ (vi) let Q(z) as in (vi). Since I := {lavll/lvl : VEE} is an internal set with I C fin *(V there exists k E IN such that I is contained in the interval *[0, k]. Hence CQ < 00

and HQ = 1jCQ > O. By Theorem 2.2 Q(z) is finite for z i:::! w. Hence (iii) ~ (ii) shows that Q(z) is complex micro continuous at w. The last implication (vi) ~ (iii) is easy: Let z i:::! wand r:= max{lzi - wil : i = 1, ... ,n}. Then w + (r, ... ,r) i:::! w and IQ(z)1 $l::VEE lavl(r, ... , r)" = Q(w + (r, .. " r)) i:::! Q(w) .•

2.4 Lemma: Let e E fin *(Vn and ht: *(Vn --+ *(V be defined by ht(v) := (el - vt) . . .. . (en - vn). Then ht is complex microcontinuous at each v E (Vn.

Proof: Let u i:::! v E (Vn. Then (ei - u;) i:::! (ei - Vi). Since (ei - Ui) E fin *(V we obtain

ht(u) = rr~=I(ei - Ui) i:::! rr~=I(ei - Vi) = ht(v) .•


2.5 Theorem: Let Q( z) = EVEE av( z - w)V be a nonstandard polynomial with a,1-' E fin *<D for all f.J, E En INn. Then

HQ(w) = sup{r E R+ : Q is finite on *Pr(w)}. (12)

Proof: Let Rq(w) be the number on the right hand side of Eq. 12. Let 0 < r < HQ(w). By Theorem 2.2 Q(z) is finite for all z E *Pr(w). Hence HQ(w) :::; Rq(w). For the other inequality let 0 < r < Rq (w). It is easy to see that Q( z) is finite on *Pr(w). Similar to the proof of Theorem 2.3 (i) => (iv) it follows that lavlr ivi =< Q(z),h.,(z) > is less than or equal to some c E R+. Hence lavll/lvl . r :::; c1/ 1vl ~ 1 for vEE with Ivl E ·lNco . Therefore st lavll/lvl :::; l/r and r :::; l/CQ(w) = HQ(w). This implies RQ(w) :::; HQ(w). The proof is complete .•

Let Q(z) = EkEEak(Z - w)k be a nonstandard polynomial in one variable with w E <D. By Theorem 2.2 HQ(w) is always lower equal to the convergence radius of the induced power series EkEEnlN st ak( z - w)k. The example Q( z) = zN with N E *lNco shows that this inequality may be strict.

3 Truncations of power series

3.1 Definition. Let P(z) = EvElNn avzv be a (formal) power series. A hyper finite truncation of P( z) is a nonstandard polynomial Q( z) of the form EVEE * avzv where E is a hyperfinite set with IW C E C *lNn . A truncation of P(z) is a hyperfinite truncation with the property that E is of the form {v E *lNn : v :::; O'} or {v E *lNn : v :::; 0', V :f:. O'} with 0' E *IN~.

3.2 Proposition: The domain of convergence of a power series P(z) is contained in SQ for every hyper finite truncation Q(z} of P(z).

Proof: Let w E <Dn be in the domain of convergence Cpo Then there exists an open polydisc Pr(O) (r E R+) with w E Pr(O) C Cpo Let Q(z) = EVEE *avzv be a hyperfinite truncation. Choose 0' E *IN~ such that E C {v E 'INn : v :::; O'}. We know that Qp(z) := Ev~p avzv converges absolutely and uniformly on Pr(O) to P(z). Hence 'Q,,(z) ~ 'Q,,(w) for z ~ w. It follows that IQ(z) - Q(w)1 :::; IQ(z)- 'Q,,(z)I+1 *Q,,(z)- ·Q,,(w)I+I·Q,,(w)-Q(w)l. Moreover IQ(z)- 'Q,,(z)1 :::; 2 Ev~",v~lNn I *avllzvl ~ 0 and a similar formula is valid for w. Hence IQ(z)-Q(w)1 ~ O .•

3.3 Theorem: Let P(z) = Evavzv be a power series. Then the following statements are equivalent:

(i) The domain of convergence contains O.

(ii) Every hyper finite truncation is complex microcontinuous at O.

102 H. Render

(iii) There exists a hyperJinite truncation which is complex microcontinuous at O.

Proof: Proposition 3.2 shows (i) =} (ii) and (ii) =} (iii) is trivial. Now let Q( z) = L:"EE *a"z" be a hyperfinite truncation. By Theorem 2.3 (ii) =} (v) we know that l·a"II/I"1 E fin·O:: for all /I E E. By a saturation argument (cf. the proof of 2.3 (v) =} (vi)) there exists B E IN with E C {/I : la"II/I"1 ~ B}. Since INn C E it follows that lal-lil/II-Il ~ B for all J.L E INn. Now a routine arument shows that the power series converges for all z E O::R with IZi I < l/B for i = 1, ... ,n: I L:I-I al-lzl-ll ~ L:I-l(lal-lll/II-II)II-Ii!zl-ll ~ L:I-I(B. r)11-I1 with r:= max{lzII, ... , IZnl} .•

3.4 Proposition: Let P( z) = L:~o akzk be a power series in one variable. If there exists w E 0:: such that Q(w) E fin·O:: for all truncations Q(z) of P(z) then P(z) converges for all z E 0:: with Izl < Iwl·

Proof: Let N E ·lNoo and QN(Z) = L:f=o ·akzk. Then aNwN = QN(W)-QN-I(W) E fin .0::. Trivially we have akwk E fin·O:: for k E IN. Hence the following statement is true: (3M E ·1N)('v'N E ·1N)(laNwNI ~ M). By transfer there exists M E IN with lakwkl ~ M for all k E IN. Hence IP(z)1 ::; L:~o lakllzkl < 00 .•

3.5 Corollary: Let P(z) = L:~o akzk be a power series in one variable. Then the domain of convergence coincides with the intersection of all SQN with N E ·lNoo

and QN = L:f=o ·akzk.

C. Impens has given in 3 an example of a truncated nonstandard polynomial PN

such that SPN is strictly larger than the domain of convergence. We use his example to show that SPN may not even be connected.

3.6 Example. Let cEO:: and Qi := z2'-I(c + ~y-l for i = 0,1,2, ... and let w = z(c + z). Define PN(Z) = L:!o Qi(Z) = L:i=O W2'-1 with N E *lNoo • Then

2N I 2(N-l)_1 k 2N I IwI2(N-l)_1 IPN(w)1 ~ Iwl - - L:k=O Iwl = Iwl - -Iwl-l . Hence we have for

Iwl > 1 the estimation RN(w) := lP~l;l~lw~)I) ~ Iwlj;-l (Iwl - 1) - 1 + IwI2(}..,-1)'

It follows that RN(W) i fin *0:: for all w E 0:: with Iwl > 1 and the same holds for IPN(w)l. Hence SPN is equal to the set {z EO::: Iz(c + z)1 < I} and PN is a truncation of the power series L:i Qi(Z), cf. 3. For c = -3 it is easy to see that SPN

is contained in BI(O) and BI (3) and that 0 and 3 E SPN'

3.7 Definition. Let X be a topological space and (lk)kEIN be a sequence of continuous complex-valued functions. Then the set of local convergence C((Jk)k) is the set of all x E X such that there exists a neighborhood U of x with the property that (Ik)k converges uniformly on U to some function g defined on U, or equivalently, that (Ik}k is a Cauchy sequence with respect to uniform convergence on U.

3.8 Theorem: Let U be an open subset of o::n and (Jk)k be a sequence in H(U). Then u (13)

(gj)j subsequence of (fkh


Proof: For the inclusion part let N E ·INco and define g(w) := st·fN(·W) for w E S. IN' Since 9 is continuous it follows that • fN(Z) ~ ·g(z) for all z ~ w E S. IN' As S • IN is open there exists a sequence of compact sets Kr such that Kr is contained in the interior of Kr+1 and U;:';l Kr = S. IN' Hence the following statement is true for any natural number nr:

By Eq. 14 and the transfer principle we can construct inductively a subsequence (fnr)r such that Ifnr+'(x) - g(x)1 < l/nr for all x E Knr . For x E S'IN there exists l E IN and nr > l such that x is in the interior of (Kd C Knr . It follows that (fnr)r converges uniformly to 9 on the neighborhood Knr of x. Conversely, let wE C((fnr)r) for some subsequence (fn.)r converging uniformly to some function 9 on a neighorhood U of w. Then • fnR is complex micro continuous at w for R E ·INco

and w E S'lnR' •

4 Polynomial approximation.

In this section we consider only functions in one complex variable. An internal function f: .([) -+ .([) is called real microcontinuous at w E 1R if x ~ wand x E *1R implies Q(x) ~ Q(w). If M is a subset of .([) we call Q to be M-continuous if z ~ w with z, wE M implies Q(z) ~ Q(w).

4.1 Proposition: Let Q(z) be a nonstandard polynomial in one variable. Then SQ is simply connected.

Proof: We show that ([) \ SQ does not possess a bounded arc component B: let T [0, 1] -+ SQ be a closed path surrounding B and SPI := h(t) : t E [0, I]}. For every Zo E B and z ~ Zo we have IQ(z)1 ~ max..,E 'SP'Y IQ(w)1 = IQ(wo)1 E fin *([) for a suitable Wo E SQ. Hence Zo E SQ and SQ is simply connected (but not necessarily connected). •

4.2 Theorem: Let U C ([) be an open simply connected set and f: ·U -+ .([) be ·-holomorphic. Then there exists a nonstandard polynomial Q(z) such that f(z) ~ Q(z) for all z E ns ·U.

Proof: Let P := P( ([)) be the set of all polynomials in one variable. The Runge approximation theorem [[9], p. 290] shows that the following statement is true:

I ("If E H(U))(VK E k)(Vn E IN)(3Q E P)(Vz E K)(lf(z) - Q(z)1 < -). (15)

n

Here k denotes the set of all compact subsets of U. Since U is locally compact we have ns·U = UKEk· K =: cpt ·U. Choose K E ·k with ns·U = cpt·U c K and

104 H. Render

an infinitely large N E "IN. The Transfer principle applied to the above equation completes the proof. •

4.3 Theorem: Let U c <V be an open simply connected set and f: U -t <V holomorphic. Then there exists a nonstandard polynomial Q(z) with Q(z) ~ "f(z) for all z E ns" U and SQ = U.

Proof: Let Un be a sequence of open, simply connected and relatively compact sets such that U = U;:'=lUn and Kn := Un C Un+l' For every finite subset E C <V \ U define fE,n: Kn U E -t <V by fE,n(Z) = f(z) for z E Kn and fE,n(Z) = n for z E E. Choose r E 1R+ such that Br(z) n Kn+l = 0 for all z E E. Then UE,n := Un+l U UzEEBr(z) is an open simply connected set and fE,n extends in an obvious manner to a holomorphic function on U E,n' By Runge s approximation theorem the following statement is true: for all n E IN and for every finite subset E C <V \ U there exists Q E P(<V) such that IQ(z) - fE,n(Z) I < ~ for all z E Kn U E. Choose N E "lNco and E hyperfinite with <V \ U C E c "(<V \ U). Then there exists Q E * P(<V) with "fE,N(Z) ~ Q(z) for all z E * KN U E. For every z E <V \ U C E we have Q(z) ~ N, hence SQ C U. The other inclusion is clear since ns *U = U;:'=l "Kn C "KN and Q(z) ~ "fE,N(Z) = "f(z) for all z E "KN .•

Now let us consider polynomial approximation on compact sets. Observe that the nonstandard polynomial Q in the following theorem is necessarily "K -continuous since f is continuous.

4.4 Theorem: Let K C <V be compact and <V \ K connected. For every continuous function f: K -t <V which is holomorphic in the interior there exists a nonstandard polynomial Q(z) such that Q(z) ~ "f(z) for all z E "K and Q(" zo) fj fin *<V for all Zo E <V \ K. In particular, if K is nowhere dense then SQ is void.

Proof: Let D := <V \ K. For every finite subset E C D we define a continuous function IE: K U E -t <V by fE(Z) = f(z) for all z E K and fE(Z) = lEI for z E E where lEI denotes the cardinality of E. Now KE := KUE is compact with the same interior as K and its complement is still connected. By the Theorem of Mergelyan [[9], p. 423J the following statement is true: for every fE, for every n E IN there exists Q E P(<V) such that IQ(z) - fE(z)1 < ~ for all z EKE. Choose E hyperfinite with <V \ K c E c *(<V \ K) and N E "lNco and use the transfer principle .•

It is also possible to construct a nonstandard polynomial Q(z) with Q(z) ~ * f(z) for all z E "K such that (i) Q(zo) E fin "<V for all Zo E <V \ K and (ii) Q is nowhere complex micro continuous on <V \ K. For example, divide <V \ K into two disjoint dense set D1, D2 and define fE(Z) = i for z E EnD; with i = 1,2. Moreover one can achieve that IE restricted to any line segment I C <V \ K is not" I-continuous, choose Dl := D n (R \ IQ X IQ) and D2 = D \ D1• Here IQ denotes the set of all rational numbers.

4.5 Example. Let f be a holomorphic function on the open unit disk Bl such that (i) f can be extended continuously to the closed disk D and (ii) the partial sums


sn := 2:;=0 ak are not bounded where f(z) = 2:~0 anzn for all z E Bt, cf. [[6], p. 30]. By Theorem 4.4 there exists Q(z) = 2::=0 bkzk with Q(z) ~ * f(z) for all z E * D. Now Theorem 2.2 shows that

N 00

st (L bkzk) = L st bkst zk for all z E ns * B1• (16) k=O k=O

Hence we have st bk = ak for every k E IN. Although the left side is meaningful for every z E * D the right side does not exist for z = 1.

We remark that the validity of Eq. 2 only for all z E Pr(w) (instead of *Pr(w)) is not equivalent to complex microcontinuity as our next example shows.

4.6 Example. Let E := {Zl,"" ZN} be a hyperfinite set with Pr(w) C E and let P(z) := rr:=l(Z - Zi). Choose y ~ w with y ~ E and choose c E *JR such that \c(y - W)N P(y)\ is infinitely large. Then Q(z) := c(z - w)N P(z) is not complex micro continuous since Q(y) is infinitely large and Q( w) = O. On the other side we have Q(Zi) = 0 for all Zi E E.

Let U be an open subset of <V and choose a sequence of open relatively compact subsets Un with Kn := Un C Un+1 and U = U~=lKn' Endow the set C(U) of all continuous complex-valued functions with the locally convex topology induced by the norms \\fn\\ := sUPzEKn \f(z)\ (n E IN). Choose an internal subspace E of *C(U) such that P(<V) C E C *P(<V). Call Q E E finite if \\Qn\\ E fin*JR for all n E IN. The set of all finite points of E is denoted by fin E. It is obvious that finE = {Q E E: (Vz E ns*U) (j(z) E fin *<v)}. Now the equivalence of (iii) and (i) in Theorem 2.3 shows that

fin E = {Q E E: Q is microcontinuous at every z E U and Q(z) E fin *<v}.

Hence the standard part st is a (linear) map from E into H(U). If U is simply connected then it is also surjective: For f E H(U) apply the overflow principle to the set {n E *IN : (3Q E E)(Vz E *Kn)(\*f(z) - Q(z)\ < ~). In this case the nonstandard hull E of E (for definition see [1]) coincides with H(U). In the general case E is isomorphic to the algebra of all holomorphic functions on U which are limits of polynomials with respect to compact convergence in U.

References

[1] C.W. Henson, L.C. Moore, The nonstandard theory of topological vector spaces. Trans. Amer. Math. Soc. 172 (1972) 405-435.

[2] C. Impens, Local microcontinuity of nonstandard polynomials. Israel J. Math. 59 (1987) 81-97.

[3] -, Propagation of microcontinuity for nonstandard polynomials. Journal D' Analyse Mathematique 53 (1989) 187-200.

106 H. Render

[4] - , Real functions as traces of infinite polynomials. Math. Ann. 284 (1989) 63-73.

[5] -, Standard and nonstandard polynomial approximation. To appear in J. Math. An. Appl.

[6] E. Landau, D. Gaier, Darstellung und Begriindung einiger neuerer Ergebnisse der Funktionentheorie. Springer Berlin 1986.

[7] L. Nachbin, Holomorphic functions, domain of Holomorphy and local porperties. North-Holland, Amsterdam 1972, Second Edition.

[8] A. Robinson, Non-Standard Analysis. North-Holland, Amsterdam 1966.

[9] W. Rudin, Real and complex analysis. McGraw-Hill, New Dehli 1974.

AN EXISTENCE RESULT FOR A CLASS OF PARTIAL DIFFERENTIAL EQUATIONS WITH SMOOTH

COEFFICIENTS

Todor Todorov Mathematics Department, California Polytechnic State University,

San Luis Obispo, California 93407, USA

Abstract

We prove the existence of a solution for a class of linear partial differential equations with smooth coefficients in a space of generalized functions larger than the space of Schwartz distributions. As an example, we show that H. Lewy's equation has a solution in this space whenever its right hand side is a classical smooth function or a Schwartz distribution. AMS subject classification: 03H05, 30GIO, 35A05, 35A08, 46FlO. Keywords: differential operator, fundamental solution, generalized solution, partial differential equation, Schwartz distribution, nonstand analysis, internal function, transfer principle, saturation principle.

o Introduction

The main purpose of this paper is to show that the equations of the type:

P(x,8)U = F (0.1)

have solutions U in '£ for any choice of the right hand side F also in £. Here '£ is a class of generalized distributions, larger than the class of Schwartz distributions VI, defined in Section 2 in the framework of nonstandard analysis (Tom Lindstr(llm [8]). The result is proved for a class of linear partial differential operators with Coo -coefficents:

P(x,8) = L c,.(x)8", (0.2) l"l~m

called in this paper "regular" (Section 1). As a consequence, we show that H. Lewy's [7] equation in IR3:

8U .8U 2·( . )8U F - + z- - Z Xl + ZX2 - = 8Xl 8X2 8X3

(0.3)

107

108 T. Todorov

has a solution U in £ for any choice of Fin £, in particular, whenever F is a classical smooth function or a Schwartz distribution. In addition, we show that the class of generalized distributions £ preserves the solvability of the linear partial differential equations with constant coefficients

P(8)U = F (0.4)

in the sense that the equation (0.4) has a solution U in £ for any choice of Fin £.

We denote the usual classes of the Goo-functions and Goo-functions with compact supports defined on IRd (d is a natural number) by: £ = £(lRd) = Goo(lRd) and V = V(lRd ) = Go (lRd ), respectively, and by £' = £1(lRd) and V' = V'(lRd ) we denote the corresponding classes of Schwartz distributions on IRd (L. Schwartz [11]). For integration over IRd we use the Lebesgue integral. As usual, IN, IR and <V will be the systems of the natural, real and complex numbers, respectively, and we use also the notation lNo = {O} U IN. For the partial derivatives we write 8"', a E INg. If a = (all a2,"" ad) for some a E lNg, we write lal = a1 + a2 + ... + ad. If f is a function or Schwartz distribution, we denote j(x) = f(-x). Finally, we denote by £ == £(lRd) the class of "generalized distributions on IRn " , defined in Section 2.

Recall that any linear partial differential equation with constant coefficients (0.4) has a solution U in V' for any choice of F also in V' (L. Ehrenpreis [5J - B. Malgrance [9]). A general existence result for the linear partial differential equations with Goo-coefficients was first conjectured, then proved to be false in the settings of distributions (H. Lewy [7]) and hyperfunctions (P. Schapira [10]). In particular, H. Lewy's equation (0.3) is famous for not having solutions in V' for a large choice of F even in V. This explains why we are looking for solutions in classes of generalized functions larger than the class of Schwartz distributions 1)'.

The result presented here is related to F. Treves's works [14J, [15], where the local existence and uniqueness of the Cauchy problem for the equation (0.1) is proved for operators P(x,8) with analytic coefficients, where the right hand side, the Cauchy data and the solution are, in general, analytic functionals. For comparison we mention the following: a) Our solutions are global, whereas the analytic functionals present, in general, local solutions. b) The class of generalized £ contains the class of Schwartz distributions V' while the class of analytic functionals contains neither V', nor the class of the classical smooth functions £. Hence, in addition to the restriction on the coefficients, F. Treves's result does not include the case when the right hand side of the equation F is in £(11) - V(11) (or in V'(11) - £1(11)). e) The generalized distributions are localizable (e.g. we can define the concept of "support" of a given generalized distribution), whereas analytic functionals are not.

d) The classes of operators P(x,8) in these two different approaches are not comparable (neither of them is larger than the other), although, their intersection is comparatively large: it contains, for example, all elliptic P(x,8) with analytic coefficients.

An existence result for a class of PDE 109

Our results are related also to J.F. Colombeau's work [2), where a general existence result for the linear PDEs with smooth coefficients:

ph(x,8)U(x) rv F(x) (0.5)

has been established in the class of the "new generalized functions" 9(0) (J. F. Columbeau [1)). Here ph(x,8) is a regularization of the original operator P(x, 8) (depending on a function h) and rv is an equivalence relation in 9 (0) called "association". Later this result was improved in (J.F. Colombeau, A. Heibig, M. Obergruggenberger [3)), where the uniqueness of the Cauchy problem for the equation (0.5) was proved in the class of generalized functions 99 (0) ("g" stands for "global") and the association'" in (0.5) was replaced by the strict equality in 9}(0). In our approach the solutions satisfy the equations in the usual sense, i.e. in the sense of the equality in £ and the operator in the equation is an extension of the original operator P(x, 8) for £ onto £.

1 Regular differential operators with COO-coefficients

We specify and study a class of differential operators, here called "regular". The main result of this section is presented in Lemma (1. 7).

(1.1) Definition: Let P(x,8) be a linear partial differential operator with Coo-coefficients (see (0.2) at the beginning of the Introduction). We say that P(x,8) is "regular" if its transposed operator pt(x, 8), defined by

pt(x,8)cp(x) = L (-1)1"18"[ca (x)cp(x)), 1"15m

is injective on V (i.e. if cp E V and pt(x, 8)cp = 0 implies cp = 0)

(1.2) Examples

1. The linear partial differential operators with constant coefficients are regular. To show this assume that P( 8)cp = 0 for some cp E V and take the Fourier transform :F. We obtain P(-iz)F(cp)(z) = 0 which implies F(cp) = 0 and hence cp = O. Now the result follows since pt (8) is also a differential operator with constant coefficients.

2. The elliptic linear partial differential operators with analytic (entire) coefficients are regular. This follows immediately from (L. Hormander [6], Theorem (7.5.6), p. 178) and the fact that the transposed operators are also elliptic with analytic coefficients.

3. The second-order elliptic linear partial differential operators with Coo -coefficients are regular.

110 T. Todorov

4. H. Lewy's operator:

in lR3 (H. Lewy [7]) is regular. This follows from the following result (also due to H. Lewy): Let us imbed lR3 in <v2 by the mapping .

(Xl. X2, X3) -t (Xl + iX2, x3 + i(xi + xm.

Then lR3 becomes the boundary of the domain D = {Imz2 > IZI12}, and any solution of Lu = 0 extends holomorphically into D. Hence we get u = 0, by the unique continuation for holomorphic functions. Now the result follows since V(x,8) = -L(x,8).

5. By contrast with the above, the operator P(x,8) = '!/;8Q is not regular for '!/; E 1), '!/; :f. 0, since pt(x, 8)cp = 0 for any cp E 1) such that supp'!/; n suppcp = 0. 6. The operator P(x, 8) = xI8/8x2-X28/8xI in lR2 is not regular, since P(x, 8)cp(xi+ x~) = 0 for any cp E 1)(IR) and pt(x, 8) = -P(x,8).

(1.3) Notation: Let P(x,8) be a fixed linear partial differential operator with Coo-coefficients. For any cp E 1) we define the functional P", : 1) -t <V by

P",(u) = J [P(x,8)u)(x)cp(x)dx,

lRd

We write ker P", for the kernel of P",.

Notice that the integrating by parts we get

u E 1).

P",(u) = J u(x)[pt(x,8)cp](x)dx

lRd

(1.4)

for all u E 1). Hence P", = pt(x, 8)cp where pt(x, 8)cp is considered as an element of 1)' (in the sense of Schwartz theory of distributions). It follows, in particular, that P", E 1)' for any cp E 1).

(1.5) Lemma: Let P(x, 8) be a linear partial differential operator with Coo-coefficients. Then P(x, 8) is regular iff the mapping cp -t P", from 1) into 1)' is injective.

Proof: The result follows immediately from P", = pt(x, 8)cp .•

In what follows, we shall work in a nonstandard model of the set of the complex numbers <V with degree of saturation larger than card lR. In particular, any polysaturated model of <V will suffice (Tom Lindstr!llm [8], p. 49). If X is a set of complex numbers or a set of (standard) functions, then * X will be its nonstandard extension and if f : X -t Y is a (standard) mapping, then * f : * X -t *y will be its nonstandard extension. We shall use the same notation, *, for the convolution


operator * : V' X V --+ e and its nonstandard extensions * : ·V' X *V --+ *e. For integration over *IRd we use the *-Lebesgue integral (Tom Lindstr(6m [8], Ch. I). We shall systematically apply the saturation and transfer principles in the form presented in (Tom Lindstr(6m [8], Chapter III and Chapter IV) where the reader can find more references to the subject.

In particular for the nonstandard extension * P<p : *V --+ / =* <D of the functional P<p defined in (1.3) we have

*P<p(u) =! [*P(x,8)u]*cp(x)dx 'IRd

u E *V (1.6)

where * P(x, 8) is the nonstandard extension of the operator P(x,8) and *cp is the nonstandard extension of cpo Finally, we should mention that P<p( u) = * P<p(*u) for all u in V, where *u is the nonstandard extension of u.

(1. 7) Lemma: Let P(x,8) be a regular linear partial differential operator with Coo-coefficients and f be a function in *e. Then for any choice of k E IN and CPi E V, i = 1,2, ... , k, the system of equations:

·P<p;(u) = / f(x)*cp;(x)dx, i = 1,2, ... , k, (1.8)

'IRd

has a solution u in *1).

Proof: By induction, consider first the case k = 1 of one equation:

·P<p(u) = / f(x)*cp(x)dx (1.9)

'IRd

If cp = 0, then any u in *V is a solution of (1.9). If cp =F 0, then the set ~ = V-ker P<p is non-empty, by Lemma (1.5), and hence the function

u = P:(~d / f(x) *cp(x)dx,

·IRd

satisfies (1.9) for any choice of U1 E ~. Assume now, that the statement is true for k -1. If CPl,"" CPk are linearly dependent in V, then (1.8) is equivalent to a system of k - 1 equations and, by assumption, has a solution. If CPt, ••• , CPk are linearly independent, then the functionals P<Pl ••• , P<Pk are linearly independent in V', by Lemma (1.5), and hence the sets

k

~i = (n ker P<pj) - ker P<pp 1=1

i = 1,2, ... ,k

112 T. Todorov

are non-empty (N. Dunford and J.T. Schwartz [4], V. 3., Lemma 10, p. 421). It is easy to verify now that the function

k ° J u = L p'U(i .) f(x) *'Pi(x)dx, i=1 'P. U, d

*IR.

is a solution of (1.8) for any choice of Ui E <I>i. The proof is complete .•

2 Generalized Distributions

In this section we define a class l of the "generalized distributions" as a factor space (a nonstandard hull) of the class of nonstandard internal smooth functions o£ (Tom Lindstr!/lm [8]). We show that l contains an isomorphic copy of the class of Schwartz distri bu tions 'V'.

We consider *£ as a differential algebra over *<V with respect to pointwise addition, multiplication and internal partial differentiation. As we already mentioned, for integration over 0lR.d we use the *-Lebesgue integral (Tom Lindstr!/lm [8], Ch. I).

(2.1) Definition (Equivalence Relation in Of): We write f ~ g for f,g E *£, if

J f(x) *'P(x)dx = J g(x) *'P(x)dx

*lR.d *lR.d

for all 'P in 'V, where *'P is the nonstandard extension of 'P.

(2.2) Lemma: (i) The equivalence relation ~ in *£ preserves addition, multiplication by scalars in *<V, partial differentiation of any standard order and multiplication by a standard function *'1jJ, where '1jJ E £. (ii) Let f, g E * £, let T be a Schwartz distribution with compact support and °T be its nonstandard extension. Then f ~ g implies *T * f ~ ·T * g.

Proof: (i) The preservation of addition and scalar multiplication is obvious. The preservation of the partial differentiation follows easily by integration by parts (and the transfer principle):

J EPf(x)·'P(x)dx = (_1)1"1 J f(x)18"'P)(x)dx =

*lR.d *lR.d

= (_1)1"1 J g(x) 1EP'P)(x)dx = J 8"g(x) °'P(x)dx,

*lR.d *lR.d

i.e. 8" f ~ EPg. The preservation of the multiplication by a standard function follows simply from the fact that .( '1jJ'P) = *'1jJ .'P.

An existence result for a class of PDE

(ii) Suppose cp E 'O. Since T * rjJ E V, we have

J (*T*f}(x)*cp(x)dx= J f(x)*(T*rjJ}(x)dx= J g(x)*(T*rjJ}(x)dx

*Rd *Rd *Rd

as required. The proof is complete .•

= J (*T*g)(x)*cp(x)dx,

*Rd

The above lemma suggests the following definition.

113

(2.3) Defininition (Generalized Distributions): (i) The elements of the factor space *£(Rd)j 9:!, denoted for short bye, will be called "generalized distributions on R d ".

We define the inclusion £ c e by the injection f -+ q(* f), where q : *£ -+ e is the quotient mapping. If ~ ~ * £, we write ~ = q[ ~]. In particular, when ~ = * S for some S ~ £, we write S (instead of the more precise :s). (ii) We supply e with addition, multiplication by scalars in *<D and partial differentiation of any standard order inherited from *£.

(iii) We define a multiplication by a function in £ by 'lj;q(f) = q(*'Ij;f) for'lj; E £ and f E *£, where *'Ij; is the nonstandard extension of 'Ij;.

(iv) We define a convolution with a Schwartz distribution with compact support by T * q(f) = q(*T * f) for T E £' and f E *£, where *T is the nonstandard extension ofT.

(v) We define the pairing between e and V with values in *<D, by

(q(f),cp) = J f(x)*cp(x)dx (2.4)

*lRd

for f E *£ and cp E V, where *cp is the nonstandard extension of cpo

The correctness of the above definitions follows immediately from Lemma (2.2). We shall show only that the mapping f -+ q(* f) from £ into e is injective: Assume that q(* f) = 0 in e, i.e.

J *f(x)*cp(x)dx = 0

*Rd

for all cp E 'O. It follows that f = 0 in £, as required since

J *f(x)*cp(x)dx = J f(x)cp(x)dx, (2.5)

*Rd Rd

by the transfer principle (Tom Lindstr!2Sm [8], Chapter IV).

114 T. Todorov,

We should mention that the pointwise multiplication in * e can not be transferred to e since the equivalence relation ~ does not, in general, preserve the multiplication in *e.

(2.6) Proposition: (i) e is a differential linear space over *<D and e is a differential linear subspace of e over the standard scalars <D. (ii) e is a module over e and the mapping f --+ q(* f) from e into e is an e - module homomorphism. (Or, equivalently, the multiplication by a function in e generalizes the usual multiplication in e in the sense that if '1/1, fEe, then 'I/1q(* f) = q(* ('1/1 f)). (iii) If FEe, then (F, cp) = 0 for all cp E V implies F = 0 in £. Moreover, the pairing between e and V is an extension of the usual pairing:

(f,cp) = J f(x)cp(x)dx

lRd

between e and V in the sense that (q(* f), cp) = (f, cp) for all f in e and cp in V.

The statements follow immediately from Lemma (2.2). In particular, (q(* f), cp) = (f, cp) follows directly from (2.5) .•

Let F be a generalized distribution, a be an open set of lRd and V(a) be the space of test functions on a. We say that "F is zero on a" and write "F = 0 on a", if (F, cp) = 0 for all cp in V(a). Exactly as in the case of the Schwartz distributions, we have F = 0 on n if and only if F = 0 on some open neighborhood of every point in a. Hence we can define the concept of "support": Let OF be the union of all open sets of lRd on which F is zero. Then the support of F in lRd is the complement of OF to lRd , i.e. supp F = lRd - OF.

Our next goal is to show that the space of Schwartz distributions V' "lies between e and e", i.e. that e C V' C £. We shall use some results of [12] and [13].

Let ~ E *V be a density (kernel) of the Dirac distribution r5 in the sense that

J ~(x)*cp(x)dx = cp(O)

·lRd

(2.7)

for all cp in V where *cp is the nonstandard extension of cpo A proof of the existence of a nonstandard function ~ with these properties can be found in [12].

(2.8) Definition: We define the inclusion V' C e by the mapping T --+ q(*T * ~) where *T is the nonstandard extension of T.

(2.9) Remark (Justification): In [15] it is shown that ifT is a Schwartz distribution, then the nonstandard function *T * ~ E *e is a density (kernel) of T in the sense that

J (*T * ~)(x) *cp(x)dx = (T, cp)

·lRd

(2.10)


for all tp in V. Hence the generalized distribution q(·T *~) does not depend on the choice of ~ which justifies the above definition.

(2.11) Lemma: Let T E 'D' and tf; E c and let tf;T be the product of tf; and T in V' in the sense of distribution theory and ·tf;(·T * ~) be the pointwise product of *tf; and·T * ~ in ·c. Then:

·(tf;T) * ~ ~ ·tf;(·T * ~).

Proof: In view of (2.10), we obtain:

/ (*(tf;T) * ~)(x) *tp(x)dx = (tf;T, tp) = (T, tf;tp) =

'IRd

= J*tf;(X)(*T*~)(X)*tp(X)dX ,

·IRd

for all tp in V, as required .•

Notice that according to the above lemma, the nonstandard function *tf;(*T *~) is a density (kernel) of the Schwartz distribution tf;T.

(2.12) Lemma * f * ~ ~ * f for all f in e, where • f is the nonstandard extension of f.

Proof: Applying (2.10) for T = f (in the sense of the distribution theory) we have

/ f(x)tp(x)dx= J(*f*~)(x)'tp(X)dX IRd IRd

for all tp in V and the result follows in view of (2.5). The proof is complete .•

(2.13) Proposition: (i) The mapping T -t q(*T * ~) from V' into i is injective and preserves the addition, multiplication by a complex (standard) number, partial differentiation of any standard order and multiplication by a standard smooth function in e. (ii) The mapping T -t q(*T * ~) from V' into i preserves the pairing between V' and V in the sense that if T E V' and tp E V, then

(T, tp) = (q(*T * ~), tp).

(iii) The mapping T -t q(*T * ~) from V' into i is an extension of the mapping f -t q(* I), from e into i (recall that e c V' in the sense of distribution theory).

(iv) The mapping T -t q(*T * ~) from V' into i preserves the "support" in V', in the sense that supp T = supp q(*T * ~), where supp T is in the sense of V' and

supp q(*T * ~) is in the sense of i.

116

Proof: (i) Assume that q(*T *.6.) = 0 in e for some T. That is

J (*T * .6.)(x) *cp(x)dx = 0

*Rd

T. Todorov

for all cp E 1), which implies T = 0 in 1)', by (2.10). The preservation of the addition, scalar multiplication and partial differentiation follows from the fact that the mapping T -+ *T * .6. from 1)' into *£ preserves those operations [13]. The preservation of the multiplication by a standard function 1/J follows directly from Lemma (2.11). (ii) follows directly from (2.10) and (iii) follows directly from Lemma (2.12). (iv) follows directly from (ii), since (T, cp) and (q(*T * .6.), cp) are both zero or non-zero. The proof is complete .•

(2.14) Examples 1. 0 -+ q(.6.) since we have *0 * .6. = .6., by the transfer principle. Hence q(.6.) represents the Dirac distribution 0 in t. 2. The generalized distributions q(.6.2 ),

q(exp(i.6.)) and q(exp(.6.)) all lie in e _1)'.

(2.15) Definition: Let P(x, 8) : £ -+ £ be a linear partial differential operator with COO-coefficients given by the symbol (0.2) (see the beginning of the Introduction). We define P(x, 8) : e -+ e by the same symbol (0.2):

P(x,8) = L c,,(x)8", 1"I:5m

where all operations are in the sense of e (Definition (2.3)).

(2.16) Lemma: Let P(x, 8) : £ -+ £ be a linear partial differential operator with COO-coefficients, PI(x,8) : 1)' -+ 1)' be its extension on V' in the sense of the distribution theory and * P( x, 8) : * £ -+ * £ be its nonstandard extension. Then: (i) For any f E *£ we have

P(x,8)[q(f)] = q(* P(x, 8)1).

(ii) The operator P(x, 8) is an extension of P(x, 8) in the sense that for any f E £ we have:

P(x,8)[q(*I)] = q(*(P(x,8)1)).

(iii) The operator P(x, 8) is an extension of PI(x, 8) in the sense that for any T E 1)'

we have: P(x, 8)q(*T *.6.) = q(*(PI(x, 8)T) * .6.).

Proof: (i) follows directly from the definition of the operations in e (Definition (2.3)) and (ii) follows simply from the fact that *(P(x,8)1) = *P(x,8)*f (by the transfer principle). (iii) We have to show that

* P(x, 8)(*T *.6.) ~ *(PI(x, 8)T) *.6..


Using (2.10) twice (and also "integrating by parts" along with the transfer principle), we get

J *P(x, 8)(*T * il)(x) *<p(x)dx = J (*T * il)(x)1Pt(x, 8)<p(x))dx

'Rd ·Rd

= (T, pt(x,8)<p(x)) = (PI(x,8)T, '1') = J (*(PI(X, 8)T) * il)(x) *'1' (x)dx ,

·Rd

for all 'I' in V, as required .•

(2.17) Simpler Notation: (i) We shall often write simply P(x, 8) instead of the more precise P(x,8) when no confusion could arise, e.g. the action of P(x, 8) in e will be written as:

P(x,8)[q(j)J = q(* P(x, 8)f), (2.18)

for f E *e. (ii) We shall sometimes use the same notation T for the original distribution T and its image q(*T * il) in e writing sometimes T E V' c e to stress the fact that T is considered as an element of e. For example, 8 E V' c e will denote the image q( il) in the class e of the original Dirac distribution 8 E V'.

3 An existence result

We obtain an existence result for a class of partial differential equations with Coo_ coefficients in the space of generalized distributions e. In particular, we prove that H. Lewy's [7J equation has a solution for any choice of the right hand side a classical smooth function or Schwartz distribution. Recall that fj = {q(j) : f E *V} is a subspace of E (Definition (2.3) - (i)).

(3.1) Theorem: Let P(x,8) be a regular linear partial differential operator with Coo-coefficients (Section 1). Then the equation:

P(x,8)U = F, (3.2)

has a solution U in fj for any choice of F in e (in particular, for any choice of F in V').

Proof: Suppose that F = q(j) for some f in * e. We show first that there exists u in 'V such that

'P(x,8)u ~ f

in "e. Or, equivalently, we have to show that

*Pcp(u) = J f(x)*<p(x)dx,

·Rd

118 T. Todorov

for all cp in V, where * Pcp is the nonstandard extension of Pcp (Section 1). Define the family of internal sets Acp, cp E V, by

Acp = {u E *'0: *Pcp{u) = J f{x)*cp{x)dx},

.1Rd

and observe that, by Lemma (1.7), it has the finite intersection property. Hence, by the saturation principle (Tom Lindstrl/lm [8], Chapter III), the intersection

A=nAcp cpE1>

is non-empty. Hence, every u in A has the desired property. Now the generalized distribution U = q{u) satisfies (3.2) since, in view of (2.18), we have

P(x, a)U = P{x, a)q{u) = q{* P{x, a)u) = q(f) = F,

as required. The proof is complete .•

(3.3) Corollary Every regular linear partial differential operator with Coo -coefficients P{x, a) has a fundamental solution E in V, i.e.

p{x,a)E = 8,

where 8 is the Dirac delta distribution in £. Proof: Let F = 8 in Theorem (3.1) .•

(3.4) Corollary: H. Lewy's equation:

au .au 2'{ . )au F - + t- - t Xl + tX2 - = aXl aX2 aX3

(3.5)

has a solution U in V{1R3) for any choice of F in E{1R3) (in particular, for any choice of F in V ' (1R3». Proof: The existence of U follows immediately from Theorem (3.1), since the operator

L{x, a) = aa + iaa - 2i{Xl + iX2)aa Xl x2 x3

is regular (Section 1) .•

Recall that H. Lewy's equation does not have solutions for a large choice of the right hand side F even in V: neither in the class of Schwartz distributions (H. Lewy [7)), nor in the class of hyperfunctions (P. Schapira [10)). If the right hand side F is in £' (in particular, in V), H. Lewy's equation (3.5) does have local solutions in the class of the analytic functionals (F. Treves [14], [15)). However, this result does not include the case when F is in V' - £1 (or in £ - V).

The next result shows that the class of generalized distributions E preserves the solvability of the linear partial differential equations with constant coefficients.


(3.6) Corollary: Let P( 8) be a linear partial differential operator with constant coefficients. Then: (i) The equation

P(8)U = F,

has a solution U in 15 for any choice of F in l. (ii) P(8) has a fundamental solution E in 15, i.e.

P(8)E = 8

(3.7)

(3.8)

(iii) If F is a Schwartz distribution with compact support, then the generalized distribution given by the formula:

U = F*E, (3.9)

is a solution of the equation (3.7) for any choice of the fundamental solution E of the operator P(8) in l (even when E is outside V'). (iv) The equality (3.9) holds for any solution U of (3.7) in c' and any fundamental solution E of the operator P( 8) in l (even when E is outside V').

Proof: (i) follows directly from Theorem (3.1), and (ii) follows directly from Corollary (3.3), since P(8) is regular (Section 1). (iii) Suppose that E = q(v) for some v E *c, i.e. *P(8)v ~~. In view of (2.18), we have

P(8)U = P(8)(F * q(v» = P(8)q(* F * v) = q(* P(8)(* F * v» =

= q(*F * *P(8)v) = q(*F *~) = F,

as desired, since * F * (* P( 8)v) ~ * F *~, by Lemma (2.2) - (ii), and q(* F * ~) = F (Notation (2.17) - (ii». (iv) U E C' implies FEc' and P(8)U = F in V'. Suppose that E = q(v) for some v E *c, i.e. *P(8)v ~~. By Lemma (2.2) -(ii), we get

*U * ~ ~ *U * (*P(8)v) = *(P(8)U) * v = *F * v

in *c. Hence it follows q(*U *~) = F * E in l which coincides with (3.9) in view of Notation (2.17) - (ii). The proof is complete .•

A Concluding Remark: We would like to stress that in our view the solvability of H. Lewy's equation in l is due to the sim£le fact that l is larger than V' rather than to some unique properties of the class c. In contrast to this, in the case of F. Treves' solutions in the class of analytic functionals, the property "to be solvable" for a given equation is connected with the property "to be localizable" for the corresponding solutions: " ... what lies at the root of nonsolvability of PDEs is our insistence in having solutions which are localizable generalized functions" (F. Treves [16], p. 571). Here it is meant that the Schwartz distributions are "localizable" but the analytic functionals are not. Notice that the generalized distributions in l are also "localizable" in exactly the same sense as the Schwartz distributions are. This

120 T. Todorov

is a simple consequence of the existence of a pairing between £ and V (Proposition (2.6), (iii)).

Acknowledgement: The author would like to thank Thomas K. Boehme, Ed Glassco, Goro Kato and Kent Morrison for the friendly critical discussion of the manuscript and also to Gerald B. Folland, Michael Oberguggenberger, Jean-Pierre Rosay and Francois Treves for their fruitful correspondence.

References

[1 1 J.F. Colombeau, "New Generalized Functions and Multiplication of Distributions", North-Holland Math. Studies 84, 1984.

[2 1 J.F. Colombeau, "New General Existence Result for Partial Differential Equations in the Coo Case", Preprint of the Universite de Bordeaux, 1984.

[3 1 J.F. Colombeau, A. Heibig, M. Oberguggenberger, "Generalized solutions to partial differential equations of evolution type", Preprint of Ecole Normale Superieure de Lyon, France, 1991.

[4 1 N. Dunford and J.T. Schwartz, "Linear Operators, Part I: General Theory", Interscience Publishers, Inc. New York, 1958.

[5 1 L. Ehrenpreis, "Solutions of some problems of division III", Am. J. Math., 78(1956), p. 685.

[6 1 L. Hormander, "Linear partial differential operators", Grund. Math. Wis., 116(1963), Springer Verlag.

[7 1 H. Lewy, "An example of a smooth linear partial differential equation without solution", Annals of Mathematics, Vol. 66, No.1, July, 1957.

[8 1 T. Lindstr~m, "An Invitation to Nonstandard Analysis" in "Nonstandard Analysis and its Applications" , edited by Nigel Cutland, London Mathematical Society Student Texts 10, Cambridge University Press, 1988.

[9 1 B. Malgrange, "Sur la propagation de la regularite des solutions des equations a coefficients constants", Bull. Math. Soc. Sci. Math. Phys. R.P. Romuanie 3 (53) (1959), 433-440.

[10 1 P. Schapira, Une equation aux derivees partielles sans solutions dans l'espace des hyperfonctions, C.R. Acad. Sci. Paris Ser. A-B 265 (1967), A665-A667. MR 36, #4112.

[11 1 L. Schwartz, "Theorie des Distributions I, II", Herman, Paris 1950, 1951.

[12 1 T. Todorov, "Nonstandard Delta Function", Proceedings of the American Mathematical Society, Volume 110, No 4, December 1990, 1143-1144.


[13 1 T. Todorov, "Pointwise Kernels of Schwartz Distributions", Proceedings of the American Mathematical Society, Vol. 114, No 3, March 1992, 817-819.

[14 1 F. Treves, "On Local Solvability of Linear Partial Differential Equations", Bulletin of the American Mathematical Society, Vol. 76, No 3, May, 1970, 552-571.

[15 1 F. Treves, "Ovcyannikov theorem and hyperdifferential operators", Notas de Matematica, no. 46. lnst. Mat. Pura. ApI. con. Nac. Pesquisas, Rio de Janeiro, 1968.

ON THE GENERATION OF TOPOLOGY BY EXTERNAL EQUIVALENCE-RELATIONS

Bernd Wietschorke Fachbereich 7 - Mathematik Berg. Univ. Wuppertal, Gau

str. 20 D-42119 Wuppertal

Abstract

Within the framework of Nelson's Internal Set Theory and the theory of external equivalence relations developed by Reeken we consider such relations in a vectorspace. In general the external set of infinitesimals is not a monad and therefore doesn't define a topology. We give a necessary and sufficient condition for the smallest monad containing this set to be the monad of a (not necessarily linear) topology. Furthermore a criterion is deduced that depends only on the defining sets of the given relation and makes no use of intrinsic information about this monad. Finally an application to a certain class of locally convex spaces is given.

Preliminary remark: All results presented in this article will become part of my thesis which will be finished in 1993. All statements or comments without proof will appear there.

The main context of this work is Nelson's Internal Set Theory (1ST) [3]. External sets in the sense of 1ST are underlined or carry an index like (J' (for the external set of all standard elements in a given set), l and so on. In general external sets are pointed out. Reeken has shown in [5] how in 1ST external quotients of standard spaces with respect to external equivalence relations define new standard spaces. The construction encompasses topological completions but is more general and transcends the topological framework. Without loss of generality we assume that the relation on a standard set X has the following form:

122

On the generation of topology by external equivalence-relations 123

Here U and V are standard directed sets and Q is a standard formula, that is directed in U and codirected in V (see [5]). Of course, the relation is assumed to be symmetric and transitve, and we define

X~:= {x E Xix ~ x} and X~ := {[x] I x E X~},

which we call the domain respectively the classes of the relation. Both are external sets in the sense of 1ST. Furthermore we assume two additional properties of the relation:

(i) s ~= idxxx , i.e. the standardization of the relation is the equality

(ii) the density condition: \Ix E X~3stu\lstv3sty EX: Q(x,y,u,v)

If these conditions are satisfied, the main result of Reeken is: There exists a standard set X~, called the shadowspace, and an external bijection

Ii.: X~ -+ X~,

where the index (J' denotes the external set of all standard elements of the given set. So the classes of the relation represent the standard elements of X~.

In the following we consider the problem, how such an external relation can be used to construct -a topology and how this topology can be described. This can be done in a rather general frame but within the limitations of this article we restrict to a special situation. Let X be a vectorspace and ~ an external relation on X which is compatible with the vectorspace-operations:

x ~ x' /\ Y ~ y' => X + Y ~ x' + y' X ~ x' /\ A ~K A' => AX ~ A' x'

Here ~K denotes the usual relation in the field K (R or C), i.e. both scalars are limited with infinitesimal distance. Then the class of a standard element x is given by [x] = x + [0]. For abbreviation let

!:!:.:= [0] = {x E Xix ~ O}.

It is well known, that in case f1, is a monad it directly induces a vectorspace-topology on X. In general f1, is an external collection, which can be represented as an external union of monads.If we denote

Quv:= {x E XIQ(x,O,u,v)},

and for u E Uu

124 B. Wietschorke

!!:. has the form:

In order to construct a topology p, has to become a monad, and the first attempt leads to the smallest monad that contains p" the so called discrete monad Q.(p,): if :F is defined as - -

:F :=5 {F ~ XI!!:. c F},

then :F is a filter and Q.(!!:.) the corresponding filter-monad:

Benninghofen, Richter and Stroyan [1] have discussed this monad with the aid of 7l"-monads. They have shown, that Q.(p,) in general does not define a topology, and they gave a necessary and sufficient condition when Q.(p,) does. In the following we present an explicit construction of the finest topology a given relation gives rise to, and we deduce a criterion which states when Q.(p,) is the monad of a topology. This criterion is not an intrinsic condition for Q.(ii) like in the mentioned article but depends only on the defining sets U and V of therelation. At first it is necessary to deal with the relationship between X and the shadowspace X"'. As already mentioned there exists an external bijection Is. : X"'- --+ X~. If r. : X"'- --+ X"'- is the external map which sends each element in X"'- to its class, we get by standardization of Is. 0 r. a map

i:X--+X"',

which is injective by (i) from above and in this special context also linear. For a standard set M ~ X we define

SXM:= 5{z E X"'13x E MnX",-: Z =Is.or.(x)}

So sXM is the standard set in the shadowspace the standard elements of which are represented by classes in M. The operation Sx will play the central role in the further context.

The method we use works with what we call the Robinson-Principle. The well known lemma of Robinson states, that for example an internal sequence, which is infinitesimal in all standard components, must stay infinitesimal up to a nonstandard component. Essential for this property is, that "infinitesimal" is described by a monad. For an arbitrary external relation there is at first no hope, that this theorem remains valid. But in some part it can be generalized and turns out to be useful. For abbreviation we put

A( J) := {M ~ JIM finite and J" ~ M}


Definition 1 If J is a standard directed set, ~ is said to satisfy the RobinsonPrinciple with respect to J if the following holds:

(Y')'EJ C X A vst~: Y. ~ 0 ~ 3M E A(J)V~ EM: Y. ~ 0

Definition 2 If J and K are directed sets, J is called larger than K if the following is true:

V<j> : K --t J 3 ~ E J V K. E K : ¢>(K.) ~ ~

This definition may appear somewhat curious, but later on an example is given. We say that the formula Q is monadic ally directed in U iff:

Then a first result is:

Proposition 1 If U is larger than J and Q is monadically directed in U, then ~ satisfies the Robinson-Principle with respect to J.

Proof Let (Y')'EJ be a directed net with the property: vst~ : Y. ~ O. This means explicitly:

vstdstuvstv : y, E QUIJ

Commutation of the first two quantifiers yields:

3st¢> E UJvstNstv : y. E Q,p(')1J

U being larger than J there is (by transfer) an u E Un with ¢>(~) ~ u for all ~ E J. Because Q is monadic ally directed we get:

"1st ~ : y. E !!:.,p(.) S;;; !!:....

Now the proposition follows by an overspill argument according to the usual RobinsonLemma .•

Remark: The reverse statement is also true if the relation satisfies some additional properties.

For the next definition let for a directed set J

J1 := {~E Jlvst~1 : ~' ~ ~}.

The elements of J1 are called large.

Definition 3 Let (X')'EJ C X be a standard net. (x,) is called ~-convergent with ~-limit x E X" if and only if:

v~ E J1 : x, ~ x

We denote this convergence by x, --t x. ~

By standardization --t defines a convergence structure on X which is compatible ~

126 B. Wietschorke

with the vectorspace operations. Sometimes this is also called a limit vectorspace.

Definition 4 A standard set A ~ X is called closed if the following holds:

(x.) ~ A 1\ x, 7 x => x E A

A standard set 0 ~ X is called open, if X \ 0 is closed.

Proposition 2 Let A ~ X be standard and yEA n X!:::!.. Then there is a standard net (Y")"EV C A and a set M E A(V) such that:

Proof The density condition for y implies:

3stuV'stv3stx EX: Q(x,y,u,v)

Commutation yields:

3stu3st 4> E XVV'st v : Q(4)(v),y,u,v)

By overspill we can find some M E A(V) with

For v E Vu arbitrary we have {z E AIQ(4)(v),z,u,v)} 1: 0, because this set contains y. Being standard, we can find some standard z in it and get by transfer:

3st1jJ E AVV'v : Q( 4>( v), 1jJ( v), u, v)

For v E MI it then follows from (*): 1jJ( v) ~ 4>( v) ~ y and the proof is finished .•

Proposition 3

(i) A standard set A is closed if and only if: (y E A, x E Xu 1\ Y ~ x) => x EA.

(ii) A standard set 0 is open if and only if: V'st x EO: x + !:!:. C 0

(iii) The open sets in the sense of definition 4 satisfy the axioms of a topology. This topology is translation-invariant and X equipped with this topology is a T1-space.

Proof

(i) Let A be closed, yEA, x E Xu with x ~ y. By proposition 2 there is a standard net (Ytl) C A and a M E A(V) such that y" ~ y for v E MI. It follows Ytl -t x, (what will not be shown here because it requires a lot of

~

quantifier manipulation) and thus x E A because A is closed. The reverse statement is trivial.


(ii) Follows immediately from (i)

(iii) That the axioms of a topology are satisfied follows immediately from (ii). Having (ii) in mind it is easy to see that 0 is an open neighborhood of x E X" if and only if x - 0 is an open neighborhood of O. So the topology is translation invariant. For x E X" 0 := X \ {x} is open not containing x. This shows that X is a T1-space .•

The last proposition shows that /.L and ~ characterize open and closed standard sets in the usual nonstandard way th-;ugh /.L is not a monad. The now established topology is called-the associated topology with respect to the relation. Property (iii) of the last theorem shows that the associated topology is determined by the neighborhood-monad of O. Let us denote .

~O(t::):= n 0 with 0:=5 {O ~ XI0 open and 0 EO}. DeOa

The question, when ~(/.L) defines a topology can now be transformed in the question: under what conditions-does ~o(/.L) = ~(/.L) hold? Before answering this question we establish some properties of clos~d sets Tn that topology.

Proposition 4 For a standard set A ~ X the following statements are equivalent:

(i) A is closed

(ii) (xv)vev ~ A t\ Xv '1 x => x E A

(iii) W1osX)(A) = A

Proof (i)=> (ii) If A is closed (ii) holds trivially by definition of "closed". (ii)=> (iii) The inclusion A ~ W1osx)(A) is always true what can easily be checked. So let x E (i-1oSx)(A)". There is some yEA such that y ~ x. By proposition 2 we find some standard net (Yv) C A /.L-converging to x. Applying (ii) we get x EA. (iii) => (i) Let x EX", yEA with y ~ x. Clearly x E (i-1osX)(A) and therefore x E A. By proposition 3 A is closed. •

If V is a countable set, (i) says that closed and sequentially closed are equivalent notions in this topology. A topological space satisfying this property is usually called sequential. The next theorem is the essential result and answers the before mentioned question.

Theorem 1 The identity ~o(t::) = §.(t::) holds if and only if

5U-1oSX) : P(X) ~ P(X)

is the topological closure-operator. This is true if and only if:

128 B. Wietschorke

Proof For arbitrary standard sets A, B

and (i-loSX)(A U B) = (ClosX)(A) U (i-loSX)(B)

is always true, so i-loSx is a topological closure operator if and only if ( *) holds. Let (*) be true. Let :F :=s {F S; XIl: C F}. Then by definition:

Q(l:) = n F FE:F~

For F E F" let OF := F \ (i-loSX)(X \ F).

We show that OF is open and contains o. For this let x E (OF)" and yEp,. If x + y ~ F we get x+y E X\F and thus because of x ~ x + y x E (i-loSx)(X \ i) in contradiction to x E OF. So we have x + y E F. The assumption x + y E (i-losx)(X \ F) yields due to x ~ x + y

what causes the same contradiction. Altogether we have shown x + y E OF, and by proposition 3 OF is open. Furthermore we have p, C F, i.e. p, n (X \ F) = 0 what implies 0 ~ (i-1oSx)(X \ F). So we conclude 0 E 0-;'. Now the first part of the theorem follows from

Qo(l:) ~ n OF ~ n F = Q(l:) ~ §.o(!:!:) FE:F~ FE:F~

Now let Qo(p,) = Q(p,) and A S; X be standard. Let x be a standard element of X \ (i-1oSx)(A). Th;n (x + p,) n A = 0. Therefore we can find some F E F" such that (x + F) n A = 0. The aSsumption gives us some open set 0 E 0" with 0 S; F, consequently (x + 0) n A = 0. Since x was arbitrarily chosen we get

i.e. AS; (i-1oSx)(A). The reverse inclusion always being true (which is not difficult to show) the proof is complete .•

The next proposition links our former results with the last one.

Proposition 5 If ~ satisfies the Robinson-Principle with respect to V then (*) holds.

Proof Let z E (i-1oSx)2(A)" for an arbitrary standard set A. We have to show z E (i-1oSX)(A). To start with there is some y E (i-1oSX)(A) nx~ such that y ~ z.


Choose a standard net (Yu) C (i-1oSx)(A) due to proposition 2 suitable for y. Then we get:

V·tv3x E A : x ~ Yu

Therefore we get some internal net (x,,) C A with the property: V·tv : x" ~ y". Now ~ satisfies the Robinson-Principle with respect to V, and so there is a v E Vi such that x" ~ y". v can be so chosen that in addition y" ~ y holds. So we get Xu ~ Y and from this z ~ x,,, i.e. z E (i-1oSx){A). This completes the proof .•

If we combine this result with proposition 1 we see, that a certain relationship between U and V implies Q.o(J.L) = Q.(J.L), i.e. we don't have to know the special form of Q.(!!). - -

Corollary 1 If U is larger than V Q.o(!:!:.) = Q.(!:!:.) holds.

In the following we deduce some remarkable properties of the topology induced by Q.o(J.L} in case Q.o(J.L) = Q.(J.L). Let ~X')'EJ be a standard-net converging to the standard element x with respect to the topology. Let A := {x,l~ E J}. By Theorem 1 we have A = (i-1osX}(A). Since x E A" there is some yEA n X,::: with x ~ y. Proposition 2 then yields a standard net (YU)"EV with y" -+ x. Therefore we have proved that the topological closure of

~

an arbitrary set B is determined by the J.L-limits of nets in B with index set V. If V is countable, topological spaces possessing this property are usually called Frechet spaces. Now let V be a countable set and (x n ), (Yn) two standard sequences converging to o with respect to the topology. We assume that (xn + Yn) does not converge to 0, i.e. there is some 0 E 0" and a subsequence (xnk + Ynk) such that:

(xnk ) being a subsequence of (xn) converges to 0 with respect to the topology. By what we have shown above there is a subsequence (xnk') with xnk . -;!t O. Then, of

1 J r

course, Ynk' tends to 0 in the topology and a further application gives a (last) subse-J

quence (Ynk' ) with Ynk' -+ O. But -+ is compatible with addition and we get from Ji li JJ. J.'

xnkj, + Ynkj; 70 a contradiction to (*). What we have shown is that convergent sequences form a vectorspace in this topology though it is not clear whether the topology is linear or not. We summarize:

Corollary 2 Let §.o(J.L} = Q.(J.L}. Then the closure of an arbitrary set A C X arises from taking J.L-limits Of arbitrary J.L-convergent nets (Xu}"EV C A. In case V is countable the set of a.ll sequences convergent in the topology is a vectorspace.

Finally we give an example which shows how the methods developed so far can be applied.

130 B. Wietscborke

Let X be a locally convex space with neighborhood base U and X' its topological dual. The space X is said to satisfy the countable neighborhood property (short c.n.p.), if the following holds: For any set of countably many neighborhoods Ul, U2, .. . exist positive scalars Pl> P2, ... such that

is again a neighborhood of o. Consider the following external equivalence relation on X':

The required properties for ~ are easily checked. The class of 0, i.e. !:!.' is given by:

Proposition 6 Let X satisfy the c.n.p. Then the associated topology to ~ on X' is given by §.(p,). X' equipped with this topology has therefore all properties mentioned in corollary 2.

Proof We want to apply corollary 1, so we have to show that U is larger than N and that the formula in (t) is monadic ally directed in U. So at first we have the freedom of defining a suitable quasiordering (in the sense of Reeken [5]) that makes U larger than N. The demand for monadical direction restricts this freedom and makes corollary 1 meaningful. We define

U S V :{::::::> 3n EN: V C nU.

"s" is a quasiordering (i.e. a partial ordering that lacks the property of antisymmetry). We show (U, S) is larger than N: For this let ifJ : N --+ U be arbitrarily given. Due to the c.n.p. we find scalars (Pn) such that

U:= n PnifJ(n} nEN

is a neighborhood of o. For a given n E N let an be an integer with Pn S an. Then it follows U ~ anifJ(n} which means ifJ(n} S U by definition of "S". Since nand ifJ were arbitrary (U, S) is larger than N. Now let U S V, both standard. We have to show i!:..u ~ !:!.v· f E i!:..u means: Vsfn: I flU I s ~. There is a standard integer m with V ~ mU and this yields immediately vstn : Ifvl S ~, i.e. f E !:!.v· •

Remark: Denote X~ := {J E X'I f bounded on U}. It turns out that §.o(p,} is the finest topology on X' that coincides with the canonical norm topology on eoch X~. This topology has been studied for example in Jarchow [2J, where it is denoted by


1Jt (X', X). One class of spaces satisfying the c.n.p. are the so called gDF-spaces (see Jarchow [2]). It can be shown that for these spaces (in fact even for a weaker class, the so called df-spaces) §o(p.) is the monad of a locally convex topology. It seems to be an open question if this is true for all spaces with the c.n.p. An example of a space with c.n.p. which is not a gDF-space can be found in Perez Carreras/ Bonet [4].

References

[1] Benninghofen B., Richter M.M., Stroyan K.D.: Superinjinitesimals in topology and functional analysis, London Math. Soc. Proc. 59, 1989, p.153-181

[2] Jarchow H.: Locally convex spaces, B.G.Teubner,1981

[3] Nelson E.: Internal Set Theory, Bull. Math. Soc., Vol. 83 , Nr.6, Nov.77, p.1l65-1198

[4] Perez Carreras P., Bonet J.: Barrelled Locally Convex Spaces, North Holland 131

[5] Reeken M.: On external constructions in Internal Set Theory, Expositiones Math. 10 (1992), p.193-247

NONSTANDARD HULLS OF LEBESGUE-BOCHNER SPACES

G. Beate Zimmer Dept. Mathematics, University of Illinois at Urbana-Champaign

Urbana, 11. 61 801, USA

Abstract

The problem of finding a representation of the nonstandard hull of Lp (ft, E) for 1 ~ p < 00 (E a Banach space) was posed by Henson and Moore in 1983 (see [3]). A related question was, whether this nonstandard hull can be described in some smooth way in terms of only Lp(ft) and E. We show that the answer to this question is negative. The nonstandard hull L;(ft, E) contains Lp(jl, E) isometrically (where jl is the Loeb-measure obtained from ft). Between Lp(jl, E) and Lp(ft, E) there is a space isometric to the space Mp(jl, E) of E-valued, not necessarily measurable functions that have an SLP-integrable ft-lifting. The Bochner integral extends in a natural way to Mp(jl, E) and Mp(jl, E) is a Banach space under the ~ norm. For an infinite dimensional Banach space E, the inclusions Lp(jl, E) ~ Mp(jl, E) ~ Lp(ft, E) are both proper. All of these observations are based on working with simple functions: the nonstandard hull of Lp(ft, E) equals the nonstandard hull of the space of simple E-valued functions.

Introduction

We use a superstructure model of nonstandard analysis and assume that the nonstandard model is at least ~-saturated, where ~ is an uncountable cardinal number. The nonstandard extension of an entity S is denoted by oS. We use ~ to denote "infinitesimally close". We write a < 00 when a is nonnegative and finite in *IR; if a is nonnegative and infinite, we set °a = +00.

The basic setup is a standard finite nonatomic measure space (X,A,ft) (we can assume it to be the unit interval with Lebesgue measure). From the nonstandard extension (*X, *A, *ft) we obtain the corresponding Loeb space (*X,L,,(*A),j?) by composing */1- with the standard part map, extending the resulting measure to the O'-algebra generated by *A, and then taking the completion. The Loeb space is again a standard finite measure space (see [4]). The nonstandard hull S of a normed

132

Nonstandard hulls of Lebesgue-Bochner spaces 133

space S is obtained by taking the elements of finite norm in "S (denoted by fin("S)) and dividing out by those of infinitesimal norm. This construction works in settings as general as a uniform space, see [2). We write 7r for the usual mapping of fin("S) onto S; we call 7r the standard part mapping.

Throughout this paper E is an arbitrary Banach space. A function 9 : X -+ E is simple if it is a finite linear combination of characteristic functions of sets in A with coefficients in E, i.e. if it can be written as

n

g(x) = L: ei lAi(X) ;=1

for n E IN, eI, ... , en E E and AI, . .. , An a collection of pairwise disjoint sets in A.

The standard concepts "jl-measurable", "Bochner integrable" and the definition of the Lebesgue-Bochner spaces Lp(jl, E) are used in the usual sense; see, for example,

[1). We denote by £;(jl, E) the nonstandard hull of the Lebesgue-Bochner space Lp(jl, E).

1 The nonstandard extension of the space of simple functions

1

On the space of simple functions the "p-norm" Ilfllp = (Ix IIfllP djl) P is actually a seminorm. Identifying functions for which the difference has norm zero (or, equivalently, which agree /.I-almost everywhere), we get a normed vector space which we will denote by Sp(/.I, E). We usually don't distinguish between a function and its equivalence class.

1. Theorem: Assume 1 ~ P < 00. Then the nonstandard hull of the space Sp(p., E) of simple E-valued functions with the p-norm is £;(/.1, E).

Proof: First note that if C is a normed space and D is a dense subspace of C then C = D. In fact, the condition that D is dense in C can be written as

"IE E IR+ "Ie E C 3d E D such that lid - ell < Eo

The *-transform of this formula asserts that for any element of "C there is an element of "D infinitesimally close. Identifying elements which are infinitesimally close in fin("C) we see that each equivalence class contains an element of fin("D), hence C ~ D. The reverse inclusion is obvious. It is well-known that the (equivalence classes of) simple functions Sp(/.I, E) form a dense subspace D of C = Lp(/.I, E) for 1 :$ p < 00, and so the previous density argument applies .•

The nonstandard extension of Sp(/.I, E) consists of all internal, "A-simple functions

134 G. B. Zimmer

c.p: *X -+ *E. Elements of *Sp(jl, E) can be written as

H

c.p(x) = L e; lA.(x) ;=1

with H E *IN, el, ... ,eH E *E and Al, ... ,AH pairwise disjoint sets in *A.

2 SLP-integrability

Definition: A function c.p : * X -+ * E is called S LP-integrable if for all H E *IN \ IN

If c.p is S LP-integrable then (by Theorem 3 of [4]) it follows that

Actually in our case this condition is equivalent to S LP-integrability because we assume that (X, fl, A) is a finite measure space.

If c.p is SLP-integrable and p > q ~ 1 then c.p is SU-integrable (apply the defintion), but the converse does not hold:

Example 1: Fix H E *IN \ IN and assume that p and q are real numbers with 1

p > q ~ 1. Then any function which has constant norm H'P on a set of measure 1/ H and is zero elsewhere is SLLintegrable, but not SLP-integrable since H!-1 :::! 0 but H~-1 = 1.

It follows from Example 1 that f.x 1ic.pIIP < 00 does not imply that c.p is SLPintegrable. This does, however, imply that c.p is SLq-integrable for 1 :::; q < p, as we now show.

2. Proposition:Let p > q ~ 1 be real numbers. Assume that c.p E *Sp(jl, E) (or at least that 11c.p11 is an internally measurable function) with f.x 11c.pIIP dfl < 00. Then c.p is S U -integrable.

Proof: We want to show that for any M E *IN \ IN

we get


Since by assumption 7 is positive and real and M E *IN \ IN, we conclude that

M9 is not finite. Since the product at the left hand side of the inequality is finite, f{lIcpllq~M} 1I<pllq d/l must be infinitesimal. •

3. Corollary: For p > q ~ I, the following implications hold for a function whose norm is internally measurable:

<P is SLP - integrable =? 1. 11<pIIP d/l < 00 =? <P is SLq - integrable. ·x

Example 1 shows that the first implication of the Corollary can not be reversed, and the following example shows that the second implication can not be reversed:

Example 2: Take any (real valued) function f E Lq(ji) \ Lp(ji). By the lifting theorem in [4] f has a SU-integrable /l-lifting <P E *Sq(/l,lR). Assume f.x 11<pIIPd/l is finite. It follows (by a truncation argument) that f.x °11<pIIP dji is finite. By the definition of a /l-lifting °11<p11 = Ilfll ji-almost everywhere, hence f.x IlfilP dji < 00,

which contradicts f ~ Lp(ji). Hence f.x 11<pIIP d/l is not finite. Multiplying any unit vector in *E by <P produces an SU-integrable function whose p-norm is infinite.

We will use Proposition 2 to describe how the nonstandard hull 4(/l, E) sits inside L;(/l, E) for p > q ~ 1. Before we do that, we need to describe the nonstandard hull 4(/l, E) and the Banach space of extended Bochner integrable functions contained in it.

3 The SLP-integrable functions

For any SLP-integrable function <p,

(3.1)

In what follows, we show that Equation (3.1) allows one to view SLP-integrable functions as E-valued functions by ignoring the set where 11<p1I is infinite. It also follows from Equation (3.1) that any SLP-integrable function in *Sp(/l, E) is necessarily an element of fin(*Sp(/l, E)).

From an SLP-integrable function <P we can obtain a function f : *X -+ E by composing <P with the standard part map wherever <p( x) E fin ( * E) and setting f( x) equal to zero elsewhere. This function f has <P as a /l-lifting, i.e. 7r 0 <P = f ji-almost everywhere. Equation (3.1) ensures that we change 11<pllp by at most an infinitesimal amount by composing <P with the standard part map.

136 G. B. Zimmer

The case p = 1 is considered in [6]: the space of equivalence classes of SLl-integrable functions is (isometrically isomorphic to) a Banach space of extended Bochner integrable functions with values in E; these functions are not necessarily measurable.

In [6] we construct a normed space from the S Ll-integrable functions in *Sl (p" E) by taking the quotient modulo elements of infinitesimal norm. This space is isometrically isomorphic to the space M(/i, E) formed from the set of equivalence classes offunctions I : "X -+ E which have an SLl-integrable p,-lifting cp E "Sl(p" E) (equivalence is modulo equality /i-almost everywhere). The space M(/i, E) is a Banach space under the norm 111111 = f.x 11111 d/i. On M(/i, E) we define an integral which generalizes the Bochner integral in a natural way as follows: we take an internal "Asimple and S Ll-integrable p,-lifting, integrate the p,-lifting internally and then apply the standard part map. The integral defined in this way shares many of the basic properties of the Bochner integral, but fails, for example, the convexity property (see [6]). The space Ll(/i, E) is in general a proper subspace of M(/i, E), and on it the extended integral coincides with the Bochner integral. If E is a Banach lattice then the space M(/i, E) is the space LH/i) introduced by Loeb and Osswald in [5], and the integral is the Loeb-Osswald integral. That is, their integral is extended by the integral introduced in [6].

For p > 1, an analogous construction works: let Mp(/i, E) be the space of (equivalence classes of) functions I : "X -+ E which have an SLP-integrable p,-lifting in "Sp(p" E). The SLP-integrability ensures a one-to-one correspondence between equivalence classes of functions and equivalence classes of p,-liftings in Sp(p" E). The integral for I is defined as for p = 1. Again we get a Banach space for the p-norm (this follows by adapting the proof of Proposition 6.6 in [5], where a lattice structure is assumed). Any function in Lp(ji, E) has a p,-lifting and the integral defined above coincides with the Bochner integral. All the proofs are similar to the proofs for p = 1 as given in [6].

Example 3: Assume E is infinite dimensional and X = [0,1]. Take an Auerbach basis {el' ... , eH} with coordinate functionals {h, ... , IH} for an internal subspace of "E with internal dimension H for some H E "IN \ IN. Thus each ei and Ii has norm 1 and J;(ej) = ° if i "I- j, while li(ei) = 1. Such an Auerbach basis was used by Henson and Moore in [3] to show that for infinite dimensional spaces E "I- E. Set Ai = [(i - 1)/ H, i/ H) for 1 ::; i ::; H. This partitions *[0,1] into H sets of internal measure 1/ H. Define I : *[0,1] -+ E by

This function has the SLP-integrable p,-lifting cp(x) = L:!l ej lA.(x), which is internal and "A-simple; therefore f E Mp(/i, E). As for any i "I- j we have 111I"(ei) -11"( ej) II 2: 1, f can not be separably valued outside a set of /i-measure zero.


4. Theorem If E is infinite dimensional, then for each p ;::: 1 the space Mp('[i, E) contains functions which are not '[i-measurable.

Proof: By the Pettis Measurability Theorem (see Theorem 11.1.2 in [1]) a function f : *X -+ E is '[i-measurable if and only if it is weakly jL-measurable and '[i-essentially separably valued. Example 3 constructs a function in Mp('[i, E) which is not '[iessentially separably valued .•

4 Functions which are not SLP-integrable

In [4] Loeb points out "in converting nonstandard integrals to standard integrals, we need to consider the fact that the product of any infinite positive number a with ~ is 1 while °a . O( ~) = +00 . 0." S LP-integrability is the concept which avoids this problem. However, the space fin(*Sp(t-t, E)) contains functions which are not S LP-integrable. For these functions

Because of this inequality, we can not view the functions in fin("Sp(t-t, E)) which are not SLP-integrable as E-valued functions.

For p > 1, any function in fin ( "Sp(t-t, E)) is, by Proposition 2, S L1-integrable and hence extended Bochner integrable in the sense of the previous section. This makes L;(t-t, E) a space of equivalence classes of extended Bochner integrable functions. The equivalence classes are determined by the p-norm, not by equality '[i-almost everywhere. The following example shows that functions in fin("Sp(t-t, E)) can agree ji-almost everywhere but differ in p-norm. Hence we can not describe functions in L;(t-t, E) in terms of E-valued functions with t-t-liftings.

Example 4: Take an e E *E of norm one and an r E fin"lR. Define

<p = He 1 [O,H-Pl and 1/; = r H e 1 [O,H-Pl'

Both functions are zero '[i-almost everywhere, but 11<pllp = 1 and 111/;llp = r.

For p = 1, the space fin ( "S 1 (t-t, E)) contains functions which are not S Ll-integrable and hence not extended Bochner integrable. Nevertheless, we can still make use of the internal integral of internal simple functions.

5. Proposition: If<p E fin(*Sl(t-t,E)), then Ix<Pdt-t E fin("E).

Proof: This follows from the internal triangle inequality:

H H H H

111 Lei lA; dt-tll = II Lei t-t(Ai) II :::; L Ileill t-t(Ai) = 111 Lei lA; II dt-t. X i=1 i=1 i=1 X i=1

Hence if 11<plh < 00 then Ix <p dt-t is finite in "E .•

138 G. B. Zimmer

Proposition 5 allows us to compose the integral with the standard part map to get an E-valued integral for functions in fin(*Sl(p"E)). If 11'1' -lPlh ~ 0 then 7r(J.x '1' dp,) = 7r(J.x lP dp,), hence the integral on §;. (p" E) is well defined.

Summary of results

For a fixed p ;::: 1, we have

Both inclusions are strict (see Examples 3 and 4). LvCf1, E) is characterized by measurability and MvCf1, E) is characterized by SLP-integrability.

For p > 1, the elements of the space £;,(p" E) can be regarded as equivalence classes of extended Bochner integrable E-valued functions. Unlike the case for the Lebesgue-Bochner spaces, however, the equivalence is not equality jl-almost everywhere, but it is given by the internal p-norm. Example 5 illustrates the difference.

On £;(p" E), a E-valued integral is defined by the composition of the internal integral with the standard part map. Due to the lack of SLI-integrability, not all functions in £; (p" E) can be regarded as E-valued functions when integrating.

For p > q;::: 1, it follows from Lemma 2 that £;,(p" E) ~ Mq(jl, E) and the inclusion is proper by example 2.

Remark: All of our constructions and definitions could have been done without excluding the case 0 < p < 1, but for the Lebesgue-Bochner spaces it is common to assume p;::: 1.

References

[ 1 1 J. Diestel, J.J. Uhl: Vector Measures, Mathematical Surveys No. 15, American Mathematical Society, Providence, Rhode Island, 1977

[ 2 1 C.W. Henson: The nonstandard hulls of a uniform space, Pacific. Jour. Math. 44, No.1, 115-137, 1972

[ 3 1 C.W. Henson, L. Moore: Nonstandard Analysis and the theory of Banach spaces, in:A. E. Hurd (ed.): Nonstandard Analysis - Recent Developments, Springer Lecture Notes in Mathematics 983, 27-112, 1983

[ 4 1 P.A. Loeb: Conversion from nonstandard measure spaces to standard measure spaces and applications to probability theory, Trans. Amer. Math. Soc. 211, 113 - 122, 1975


[5 1 P.A. Loeb, H. Osswald: Nonstandard Integration Theory in Topological Vector Lattices, preprint, 1992

[6 1 G.B. Zimmer: An extension of the Bochner integral generalizing the LoebOsswald integral, preprint, 1993

Part II:

PROBABILITY THEORY

A NONSTANDARD APPROACH TO DIFFUSIONS ON MANIFOLDS AND NONSTANDARD HEAT KERNELS

Hiroshi Akiyama Department of Applied Mathematics, Faculty of Engineering,

Shizuoka University, Hamamatsu 432, Japan

Abstract

Internal ordinary differential equations are used to construct diffusions on manifolds and nonstandard heat kernels are obtained for heat equations.

1 Introduction

This paper deals with nonstandard constructions of diffusions on manifolds and nonstandard heat kernels. We construct a Brownian motion on a compact Coo Riemannian manifold M in §2 following [Ak1], using an internal ordinary differential equation on the nonstandard extension of the bundle of orthonormal frames. Then the heat kernel of a heat equation involving the Laplacian is treated in §3 with the use of a "nonstandard delta function" on * M; the result improves [Ak1, Theorem 4.3]. In §4, we present a nonstandard heat kernel of a heat equation for sections of a vector bundle.

2 A nonstandard construction of a Brownian motion on a compact Riemannian manifold

Let ° < n E IN,1] = 1]0' with 1]0 E 'INoo:='IN -IN, and let (O,L(A),J-LL) be the Loeb space associated with the hyperfinite probability space (0 = ({ -1, 1 }'1)n, A, J-L) where A is the internal algebra of all internal subsets of ° and J-L(A) = IAI/IOI E *[0,1], A E A (see [L], [AFHL]). Each w E ° is expressed as W = (wk),wk = ±1, (k = 1,2, ... ,1]; Q = 1,2, ... , n). Let {ell e2,'" ,en} be the canonical basis of lRn (and thus of *lRn). Denote by w : '[0,1] X ° 3 (t,w) H w(t,w) E *lRn Anderson's *-random walk

n

w(t,w) = L w~(w)ea, a=l

1 (['It] ) w~(w):= r;;:. LWk+(1]t-[1]t])w~t]+1 ,

V 1] k=l

143

144 H. Akiyama

where [17t] E "Z, [17t] :::; 17t < [17t] + 1. Anderson's Brownian motion is given by b(t,w) = ~:=1 bf(w)eQ(t E [0,1]) with bf(w) = °wf(w) (the standard part of wNw)) (see [An]). Put tj = j.1.t = j /17 (j = 0,1, ... ,17); .1.jwQ = wU+1)L1t - w'fL1t, Tj = {t E

"JR: tj < t < tj+1} (j = 0,1, ... ,17 - 1).

Let (M,g) be a compact connected Coo Riemannian manifold of dimension n and 71' : O(M) --t M the bundle of orthonormal frames over M. Then O(M) is compact. We use the Riemannian connection on O(M) (cf. [KN]). For e E JRn,B(e) denotes the basic vector field on O(M) corresponding to e; that is, for r E O(M), B(e)r is the unique horizontal vector at r such that (7I'*)r(B(e)r) = re(E T".(r)M), where (7I'")r is the differential of the map 71' at r. From the map B : JRn 3 e 1-+ B(e) E X(O(M))(:= the space of Coo vector fields on O(M)), we obtain * B : *JRn 3 e 1-+ * B(e) E "(X(O(M))).

Now for each wEn, consider the *-continuous curve rt(r,w), t E "[0,1], in *(O(M)) starting from a standard point r (identified with Or) of O(M) at time t = ° and satisfying the internal ordinar y differential equation

drt = *B (dW(t,w)) . dt dt r,

(Notice that dw(t,w)/dt = ~:=1(dwNw)/dt)eQ = ~:=1(.1.jwQ/.1.t)eQ if t E T;.) Observe that "7I'(rt(r,w)),t E *[0,1], is a *-broken *-geodesic in *M (cf. [KN]). Put Xt(r,w) := o*7I'(rt(r,w)) = 71'(Ort(r,w)).

Proposition 2.1 ([AkI]). (1) (Ort(r,w)), t E [0,1], satisfies the stochastic differential equation in the Stratonovich form

n

dart = L B(e.,}or, 0 dbf, (2.1) .,=1

where odbf is the Stratonovich stochastic differential of bf.

(2) (Xt(r,w)), t E [0,1], is a Brownian motion on (M,g), that is, a !t:l.M-diffusion where D.M is the Laplacian acting on Coo functions on M.

Proof. (1) Let t E [0,1]. For every Coo function F : O(M) --t JR, we have

[qtj-1

*F(rt(r,w)) - F(r) R1 L {*Fh;+l(r,w)) - "F(rtj(r,w))} ;=0

n ['ltj-1 ['ltj-1 (1 n )

R1 ~ ~ *(B(eQ))r'j(r,w)* F· .1.jwQ(w) + ~ 2 ~ *(B(eQ))2 rdr:~rjotLlt, J

since the sum of the terms including .1.jWQ(w).1.jw.B(w) (j = 0,1, ... ,[17t]- 1) with 0: i: f3 is infinitesimal (cf. the proof of [An, Theorem 37]). Hence

n it it (1 n ) F(Ort(r)) - F(r) = ~ 0 B(eQ)Dr.(r)F·db~ + 0 2 ~(B(eQ))2 Dr.(r) F ds,

Diffusions on manifolds and nonstandard heat kernels

where ·db~ is the Ito stochastic differential of b~. The right-hand side equals L::=1 I~ B(eQhB(r)Fodb~, proving (2.1). Now (1) implies (2) (cf. [IW]) .•

Let f: M ~ lR be Coo. By Proposition 2.1, the expectation

145

u(t, x) = E[j(Xt(r))] (t E [0,1], x E M, r E 1I'-1(x)) with respect to VL is well-defined (independent of the choice of r in 11'-1 (x)) and it solves the heat equation

aau = ~~MU, u(O,.) = f, (u: [0,1] X M 3 (t,x) ~ u(t,x) inlR). (2.2) t 2

3 A nonstandard representation of the heat kernel of Eq. (2.2)

Let (M, g) be a Coo Riemannian manifold with dimM = n (0 < n E IN) and dVg the Riemannian volume density. Let COO(Mj K) be the space of K-valued Coo functions on M (K = lR or K = <C) and Co(Mj K) := {f E COO(Mj K) : the support of f is compact}.

Theorem 3.1 ([Ak2]). There exists a 'COO function ~: 'Mx'M ~ 'lR [namely, ~ E *( COO(MxMj lR))] satisfying the followin g properties:

(1) ~(x,y) = ~(y,x) = L:~=1 IPi(X)IPi(Y), (x, Y E *M), for some v E *lNoo ,IPi E *(Co(MjlR)) (i = 1,2, ... ,v).

(2) For all f E Co(Mj <C) and x E * M,

* f(x) = J ~(x, y) * f(y) *dvg(y). 'M

(3) Let T : Co(Mj <C) ~ <C be a Schwartz distribution on M, and define IT E

*(COO(Mj<C)) bYIT(Y) = *T(~(·,y)) = L:~=l*T(IPi)IPi(Y) (y E *M). Then

T(f) = J * f(y) IT(Y) *dvg(y), 'M

f E Co(Mj <C).

Proof. By the saturation principle, there exists a hyperfinite dimensional internal vector subspace V of the internal vector space *(Co(Mj lR)) over 'lR such that the external set {* f : f E Co(Mj lR)} is a subset of V. Let v = 'dimV( E *lNoo )

and take an orthonormal basis IPi (i = 1,2, ... , v) of V with respect to the internal inner product (ft, h) = I'M kh *dvg, (ft,h E V), so that for every f E Co(Mj <C) = Co(Mj lR) + A Co(Mj lR), * f is ex pressed as • f = L:~=1 Ci(* j)IPi with e;(* f) = I'M' f(y) IPi(Y) *dvg(y) E '<C. Then ~(x, y) := L:~=1 IPi(X)IPi(Y) satisfies (1), (2). Moreover, (3) also holds since I'M * hT *dvg = L:~=1 *T(IPi)e;(* f) = *T (L:~=1 Ci(* f)IPi) = *T(* f) = T(f) .•

146 H. Akiyama

Remark. When M = IRn and 9 is the Euclidean metric, T. Todorov ([T1], [T2]) proved the existence of a nonstandard delta function and (nonstandard) pointwise kernels of Schwartz distributions in a different manner.

Now assume that M is compact and connected. Let K = IR. Consider <P : [0,1] x O(M) x COO(M; IR) 3 (t, r, f) I-t E[f(Xt(r))] E IR (see §2). This <P is standard since it is identified with its graph, a subset of ([0,1] x O(M) x COO(M; IR)) x R. Since ~(.,y) E *(COO(M;IR)) for each y E *M, we obtain the internal map *[0,1] x *(O(M)) x *M 3 (t,r,y) I-t *<P(t,r,~(.,y)) E *IR.

Theorem 3.2. Let e(t,x,y) denote the heat kernel of (2.2), which is known to exist. Then *<P(t,r,~(·,y)) coincides with e(t,x,y) fort E (0,1]' x, y E M, and r E 7r-1(x).

Proof. As is well-known, e(t, x,·) is smooth for fixed (t, x). Moreover, <p(t, r, f) = E[!(Xt(r))] = IMf(y)e(t,x,y) dvg(y). Therefore it holds that *<P(t,r,~(·,y)) = I'M ~(z, y) *e(t, x, z) *dvg(z) = *e(t, x, y) = e(t, x, y) .•

4 A nonstandard heat kernel of a heat equation for sections of a vector bundle

Let P(M, G, 7rp) be.a Coo principal fiber bundle over a compact connected Coo Riemannian manifold M; in this section, we do not assume that dimM = n. Assume that G (the structure group) and P are compact. Let 7rE : E --t M be a Coo real vector bundle of rank l =f ° associated with P through a representation of G into GL(l, IR). Let EO be the dual bundle of E. The fiber 7rE/(x) of E over x E M is denoted by Ex. For x E *M, we put 'Ex := (*(7rE))-l(x).Let Ao,A1, ... ,An

be projectable Coo vector fields on P invariant under the right translation by any element of G. Let wf(w) and bf(w) be as in §2. For convenience, we put wr :=

t, LljwO := Llt.

For each wEst, consider the *-continuous curve (Pt(p, w))tE*[O,lj in • P with Po(p, w) = pEP governed by the internal ordinary differential equation

dpt = ~ dw~(w)('A ) . dt L..J dt .\ Pt

.\=0

We often write Pt(p) for Pt(p,w) by suppressing w. Each pEP with 7rp(p) = x is regarded as the admissible map IRl 3 v I-t p(v) E Ex ~ E. Observe that pt(pa) = pt(p)a, (p E P,a E G). Then for x E M, we can define Ot(x):= 7rp(Pt(p)) and Pt,x := Pt(p)o*p-1 : *Ex --t *E9t (x) independently of the choice of P E 7rpl(x). Given a E rOO (E) (:= the space of Coo sections of E), define FiT : P --t IRl by FiT(p) = p-l(a(7rp(p))), and set pt1('a)(x) = p~;(·a(Ot(x))), (x EM). We can also define L.\ : rOO(E) 3 a I-t L.\a E rOO(E) by L.\a(x) = p((A.\)pFiT)' (x E M, P E 7rp1(x), .A = 0,1, ... , n).

Diffusions on manifolds and nonstandard heat kernels 147

Proposition 4.1 (cf. [Ak1)). (1) (OPt(P))tE[0,1] satisfies the stochastic differential equation in the Stratonovich form

n

dOpt = L(Ac.h.odb~ + (Aoh. dt. a=1

(2) Put H = ! ~:=1(La)2 + Lo. Let a E rOO(E), (t, x) E [0,1) x M. Then

O(pt1(*a)(x)) - a(x) = ~ It O(p:;1(*(Laa)}(x)). db~

+ It O(p:;1(*(Ha)}(x)) ds. (4.1)

(4.2)

Proof. Part (1) is proved in a similar way as in (1) of Proposition 2.1.

(2) Since *p((* AX)p,(p)*(Fa)) = pt,";(*(L~a}(Ot(x))), we have, for t E Tj ,

:It,";(*a(Ot(x))) = *p (d(*(Fa~~Pt(P)))) = t .1~~~ pt,";(*(L~a}(Ot(x))), ~=O

Use these formulas to obtain (4.1) (cf. [Ak1)); notice that O(pt1(*a)(x)) = O(*p(*(Fa}(Pt(P)))) = p(Fa(OPt(P))) E Ex. Part (3) follows from (2) .

• Let h be a Coo fiber metric in E. By the saturation principle, there exists a hyperfinite dimensional internal vector subspace V of *(roo(E)) over *R such that {*a : a E roo(E}} ~ V. Let v = * dim V, and take an orthonormal basis 1/l; (i = 1,2, ... , v) of V with respect to the internal inner product (a1> (2)V = J'M*h(a1,a2)*dvg , (a1,a2 E V). Put D(x,y):= ~~=11/l;(X)®1/lf(y), (x,y E *M), where 1/l? E *(roo(E*)) is obtained by the evaluation map (* E: 8 * E:) ® * Ex -+ *E:, *h(x) ® 1/l;(x) ~ 1/lf(x) = *h(x) .1/l;(x), (x E *M). (For the tensor product ® and the symmetric tensor product 8, we denote *® and *8 simply by ® and 8, respectively. )

Let Ill: [0,1] x M x rOO(E) 3 (t,x,a) ~ E[O(pt1(*a}(x))] E Ex ~ E. Then III is now standard since it is identified with its graph.

148 H. Akiyama

Theorem 4.2. For t E [O,I],x E M, and y E "M, put "W(t,x,D(·,y)) :=

~f=l"W(t,x,?jI;) ® ?jI?(y) E "Ex ® tE;. Then "W(t,x,D(·,y)) is a "nonstandard heat kernel" of (4.2) as long as t E [0,1], in the sense that

E[O(p;l("a}{x))] =! "w(t,x,D(·,y))· "a(y) tdvg(Y), (t E [0, 1],x EM), 'M

regardless of whether there exists a standard (in the sense of nonstandard analysis) heat kernel of (4.2) or not.

Proof. By "a = ~f=l e;("a)?jIi with e;("a) = ("a,?jIi)V = I'M?jI?·"a "dvg, we get E[o(p;l("a)(x))] = W(t,x,a) = "w(t,x, "a) = ~f=l Ci("a) "w(t,x,?jIi) = I'M "w(t,x,D(.,y)). "a(y) "dvg(y) .•

References

[Akl] H. Akiyama, Applications of nonstandard analysis to stochastic flows and heat kernels on manifolds, in "Geometry of Manifolds" (K. Shiohama, ed.), Perspectives in Mathematics, Vol. 8, Academic Press, Boston, 1989, pp. 3-27.

[Ak2] H. Akiyama, Nonstandard representations of generalized sections of vector bundles, preprint.

[AFHL] S. Albeverio, J. E. Fenstad, R. H9legh-Krohn and T. Lindstr9lm, Nonstandard Methods in Stochas tic Analysis and Mathematical Physics, Academic Press, New York, 1986.

[An] R. M. Anderson, A non-standard representation for Brownian motion and Ito integration, Israel J. Math. 25 (1976), 15-46.

[IW] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland/K odansha, Amsterdam/Tokyo, 1989.

[KN] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, I, John Wiley & Sons (Interscience), New York, 1963.

[L] P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113-122.

[Tl] T. Todorov, A nonstandard delta function, Proc. Amer. Math. Soc. 110 (1990), 1143-1144.

[T2] T. Todorov, Pointwise kernels of Schwartz distributions, Proc. Amer. Math. Soc. 114 (1992), 817-81 9.

A NONSTANDARD ApPROACH TO THE MALLIAVIN CALCULUS

Nigel J. Cutland and Siu-Ah Ng Dept. of Pure Mathematics, University of Hull,

Cottingham Road, Hull HU 6 7RX, England

Abstract

We outline an intuitive approach to the Malliavin calculus for the classical Wiener space, showing that the basic operators of this calculus have natural descriptions as classical differential operators on a nonstandard space *lRN for an infinite natural number N.

1 Introduction

Let W denote Wiener measure on the space C = 0[0,1], and write L2(W) for the space L2(C, W) of square integrable Wiener functionals. In its simplest form, the Malliavin calculus is a kind of partial differential calculus for functionals c,o{b) in L2(W), embodying the following intuition. Each fixed sample point b E C can be thought of as given by the family of increments Xt = dbt = bt+dt - bt (t E [0,1]). Under W these increments Xt vary independently with mean 0 and variance dt. Thus, informally, we can think of c,o E L2(W) as c,o = c,o(x), where x = (Xt)tE[O,11 .

The gradient or derivation operator D is then given intuitively by

8c,o Dc,o{b, t) = -8 (b).

Xt

We will outline below how this idea can be made precise in a nonstandard framework, which allows the basic properties of D to be derived rigorously in an intuitive way, by application of the rules of classical differential calculus.

In a similar way we treat the other basic operators of the Malliavin calculus - the Skorohod integral operator 0 and the Malliavin operator L. These are given intuitively by

ou(b) = r u(b, t)dbt _ rl aua(b, t) dt 10 10 Xt

149

150 N. J. Outland and Siu-Ah Ng

for integrands u(b, t) that are not necessarily adapted. This is an extension of the Ito integral (since, informally, an adapted integrand u(b, t) depends only on (b.).<t, hence on (x.).<t, and so 8u(b, t)/8xt = 0). -

For L we have

Lcp(b) = tSDcp(b) = {I 88cp (b)dbt _ r 882~ (b)dt 10 Xt 10 Xt

which is a generalised Ornstein-Uhlenbeck operator. The nonstandard construction of Brownian motion/Wiener measure that we need for our treatment is the modification [4] of Anderson's construction [2), in which the infinitesimal random walk IlBt = ±JXt, where Ilt is a positive infinitesimal, is replaced by IlBt which is Gaussian distributed with mean 0 and variance Ilt. Details are given in Sec.2, where we also sketch a nonstandard proof of the so-called Wiener-Ito chaos decompositon theorem for L2(W) which is crucial to the standard approach to the Malliavin calculus. In subsequent sections we give our treatment of the operators D, tS and L and their fundamental properties, and indicate some of the many applications of this theory.

In this short paper we have been selective in the topics we mention, confining ourselves mostly to basics of the theory. We omit or only sketch most proofs; those we include are chosen to indicate the flavour of the nonstandard approach to the Malliavin calculus. In a subsequent paper [7) we will provide full details together with a more comprehensive treatment.

2 Wiener Measure, Ito Integrals and Chaos

The following is the construction of Brownian motion/Wiener measure given in [4); we restrict to the time interval [0,1) and to I-dimension for convenience only - our approach extends easily to d-dimensions and time t E [0,00[.

Fix an infinite N E * IN, let Ilt = N-I ~ 0 and let n = *IR.T, where T is the hyperfinite time line T = {O, Ilt, . .. ,1 - Ilt}. Elements of n are denoted by x = (Xt)teT. The internal probability r is defined on n as the product of the normal distribution N(O, Ilt) on each coordinate:

r(A) = (27l'Iltt~ { exp ( - 2~t L x;)dx 1A teT

for A E *B, the *Borel subsets of n, where dx denotes *Lebesgue measure on n. We denote by n the complete Loeb space n = (n,:F, P) obtained from (n, *B, r). Let T = T U {I}. An internal process (Bt )tet is defined on n by

A nonstandard approach to the Malliavin calculus 151

i.e.

The following was proved in [4] (following the ideas of [2]):

Theorem 2.1 For P-almost all x E n, B(x) is S-continuous, and the process bt(x) defined on n by

bot(x) = °Bt(x)

for t E T is Brownian motion, so Wiener measure is given on C by

W(A) = P({x En: b(x) E A})

Ito integration with respect to b is developed exactly as in [2] or [8]. A standard right-continuous filtration (Ft)tE[O,l) is defined on n by Ft = Nv a(Uo.=t A.), where N denotes the P-null sets and A. (s E T) is the internal ·a-algebra generated by (xu)u< •. We have

Theorem 2.2 Let f: n x [0,1] -+ lR be an adapted process with E(Jo1 Fdt) < 00.

Then

(1) f has an SL2 lifting F: n x T -+ ·lR that is nonanticipating (i.e. F(.,t) is At-measurable for each t E T);

(2) for any such lifting F, define

G(x,t) = LF(x,s)~B • • <t

and then G(x,·) is S-continuous a.s., and for a.a. x

°t

10 j(x, s)dh. = °G(x, t) all t E T.

Wiener-Ito chaos expansion

Let us fix some notation. For n ~ 1 let ~n = {(tb"" tn ) : 0 :::; tl :::; ... :::; tn :::; I}. For f E L2(~n)' the multiple Wiener integral is defined by

say. We will also write J fdb(n) and In(f) to denote this integral. The following properties of In(f) are routine:

152

Proposition 2.3 (a) E(In(f)) = 0

(b) E(In(f)2) = IIIn(f)112 = IIflit (c) if f E L2(An) and g E L2(Am) then

N. J. Outland and Siu-Ah Ng

if n# m if n= m

The nonstandard representation of the Ito integral gives the following representation for the multiple Wiener integral. The counterpart of An for the time set T is the set A~ = {(tl, ... ,tn ) E Tn : tl < t2 < ... tn }, and we have

Theorem 2.4 Let f E L2(An) with SL2 lifting F : A~ -+ oR. Then

In(f) = 0 L F{tl, ... ,tn)ABt1 ... ABtn h<h···<tn

= L F{tl, ... ,tn)Xtl ... Xtn tl <h ... <tn

It is convenient to write In{F) = Ltl<h ... <tn F{tl, ... , tn)Xtl'" Xtn. It is immediate that the counterpart of Proposition 2.3 holds for In{F), and from this Proposition 2.3 itself can be proved.

The fundamental Wiener-Ito chaos decomposition result for L2(W) is now:

Theorem 2.5 Let'P E L2(W). Then'P has a unique expression as

00

'P{b) = 'Po + L In(fn) n=l

00

11'P112 = 'P~ + L IIfnll2

where IIfnll means IIfnllAn •

We will sketch a proof of this below.

The closed subspace of L2(W) given by

n=l

is called the nth Wiener chaos; so Theorem 2.5 says that


where we set Zo = lR. Let z(n) denote E9!!.=oZm' For cP E L2{W) we write CPn for the projection of cP onto Zn and we write

n

cp(n) = E CPm

m=O

for the projection onto z(n). There are a number of proofs of Theorem 2.5 in the literature - for example [9J or [17J. In the paper [6J we gave two elementary nonstandard proofs and we sketch one of them here.

Proof of Theorem 2.5 It is well known that L2(W) has a dense subset spanned by linear combinations of functions for the form

cp(b) = exp (11 f(t)dbt - ~ 11 f2(t)dt) ,

so it is sufficient to show that cP = 1+ L'::'=1 In(fn) = 7f;(b), say, where fn(tl>'" , tn) = f(tt} .. . f(tn)' Take an SL2 lifting F of f and define an internal function ~

~(B) = II(l + F(t)LlBt). tET

The proof is completed by showing that (i) ~(B) ~ cp(b) a.s. and (ii) ~(B) ~ 7f;(b) a.s. The first follows by showing that a.s. we have

log ~(B) = Elog(l + F(t)LlBt) tET

= E(F(t)LlBt - ~F2(t)LlB; + Ct) tET

~ E(F(t)LlBt - ~F2(t)Llt) tET

~ log cp(b)

For (ii) we have the exact expansion

N

~(B) = 1 + E E F(tt}F(t2)'" F(tn)LlBtl ... LlBtn n=1 l:l.;:

where Fn(tl>." , tn) = F(tt} ... F{tn) . We have In(Fn) ~ In(fn) a.s. for all finite n, by Theorem 2.4. The observation that


which is infinitesimal for infinite n completes the proof. •

Wiener thought of the integrands fn in the chaos expansion of cP E L2(W) as being given by the following 'recipe' (see [9]):

fn(tl, ... , tn) = E( cp(b)bh bh ... btn ).

We can make sense of this as follows (see [6] for details):

Theorem 2.6 Let cp E L2(W) have chaos expansion given by Theorem 2.5, and let If? be an SL2 lifting of cpo Then for finite n the following function Fn is an SL2 lifting of fn:

(2.1)

where Bt = D.Bt! D.t.

Proof (Sketch) Take SL2 liftings Fn of fn; then for sufficiently small infinite M, the function

n=l

is an SL2 lifting of cpo It is routine to see that E(WXtlXt2'" Xtn ) = Fn(tb"" tn)D.tn and hence that IIFn - Fnll 2 ::; E((W _1f?)2) ::::; 0 for finite n .•

Suppose we now take cp E L2(W) with SL2 lifting If? and define for all n E* IN

If?n = In(Fn)

where Fn is given by the recipe (2.1), and m m

The above considerations show that

Theorem 2.7 For all infinite M, If?(M) is an S L2 lifting of cpo

Each lifting If?(M) is a monomial lifting of cp; i.e. If?(M) is a (nonstandard) polynomial in (Xt)tET with no quadratic or higher powers of any Xt. The existence of such liftings is one way to understand the Wiener-Ito chaos decomposition, and to some extent our nonstandard development shows that the Malliavin calculus on L2(W) can be seen as nonstandard classical calculus of monomials in *L2(0). However, monomials form a somewhat restricted class with few closure properties, so we do move outside this class.

Another way to understand the chaos decomposition from the nonstandard viewpoint is using Hermite polynomials. By the transfer of classical theory, any If? E *L2(0) has an expansion in terms of *Hermite polynomials (these form an orthonormal basis for *L2(0) ). For those If? that are liftings the non-monomial terms make only infinitesimal contribution.


3 The Derivation Operator

The derivation or gradient operator D is a densely defined operator

(where we take Lebesgue measure on [0,1]) with domain 102,1 ~ L2(W}, defined standardly as follows:

Definition 3.1 00

102,1 = {cp E L2(W} : L nllcpnl12 < oo} n=l

and for cp E 102,1,

where CPn = In(Jn} and in is the symmetric extension of fn to [0, l]n.

It is easy to check that IIIn- 1 (/n}11 2 = nllfnll2 (the first norm being the L2 norm with respect to WxLebesgue on C x [0,1]), and hence IIDcpll2 = 2::'=1 nllcpnll2.

For a Wiener integral cp(b}= Jo1 f(t}dbt (i.e. cp E Zl) it is straightforward to see that Dcp(b, s} = f(s} = "8cpj8(db.}" ,which accords with the intuitive description of Din the Introduction. It is not so clear in general, however, that D is a derivative. This becomes clearer from the following nonstandard approach.

For 4> E "L2(0} we have the (internal) classical derivative, the gradient V4>, given by

84> V t 4>(x} = V4>(x, t} = -8 (x)

Xt

for "differentiable 4>, and extended to general 4> by means of the "Hermite polynomial expansion of 4>. (See the Appendix for details.) In all that follows, when we write V4> or V t 4> we are assuming implicitly that 4> E dom(V} or dom(Vt } respectively.

We can now characterise 102,1 using V:

Theorem 3.2 Let cp E L2(W}. Then cp E 102,1 if and only if cp has an SL2 lifting 4> with IIV4>1I < 00.

This follows almost immediately once we have established the following lemma.

Lemma 3.3 Let cp = In(J} E Zn and let F be an SL2 lifting of f, and let 4> = In(F}. Then V4> is an SL2 lifting of Dcp.


Proof Let F be the symmetric extension of F to Tn, with F(tb .. . , tn) = 0 whenever t, = tj for some i i- j. Then F is an SL2 lifting of j and it is straightforward to check that ViP(x, s) = In - 1(F) .•

The next result is quite routine:

Theorem 3.4 Suppose that cp E ID2,1 and let iP be any SL2 lifting of cpo Then

(a) ViP(K) is an SL2 lifting of Dcp for sufficiently small infinite K.

(b) IIDcpl1 ::; ° II ViP II

(c) ViP is an SL2 lifting of Dcp if and only if IIViPII~ IIDcpli.

The following definition is useful:

Definition 3.5 Let iP E *L2 (n); then iP is SID2,1 if II V iP II is finite and II V (iP -iP(K)) II ~ 0 for all infinite K.

We have:

Theorem 3.6 Let cp E L2(W). Then

(i) cp E ID2,1 if and only if cp has an SID2,1 lifting iP;

(ii) if cp E ID2,1 with SL2 lifting iP, then ViP is an SL2 lifting of Dcp if and only if iP is SID2,1.

The next result has several applications.

Theorem 3.7 Suppose that cp E L2(W) has SL2 lifting iP, and ViP is an SL2 lifting of'I/J E L2(W X [0,1]). Then cp E ID2,1 and'I/J = Dcp (so iP is an SID2,1 lifting of cp).

We can illustrate the use of this result with the following example.

Example 3.8 Let f E L2[0, 1]; then cp = exp(lt(f)) E ID2,1 and Dcp(b, s) = cp(b)f •.

To see this, take F that is an SL2 lifting of f, so that iP = exp(I1(F)) = exp(L:tET Ftxt) is an SL2 lifting of cpo Then ViP(x, t) = iP(x)Ft and this is an SL2 lifting of cp(b)f •.

Theorem 3.7 gives an easy proof of the following chain rule for D:

Theorem 3.9 Suppose that cp E ID2,1 and f E 0 1 with bounded derivative f'. Then f 0 cp E ID2,1 and D(f 0 cp) = f'(cp)Dcp.

A nonstandard approach to the Ma1liavin calculus 157

Proof Let <11 be an SID2,1 lifting of cpo Then 1(<11) is an SL2 lifting of I(cp), and by the transfer of classical calculus, V'/(<I1} = f'(<I1)V'<I1, which is an SL2 lifting of f'(cp)Dcp .•

The chain rule for functions 1 of several variables follows in the same way.

Another easy application of Theorem 3.7 is

Theorem 3.10 D is a closed operator.

Proof Let CPn --+ cp in L2(W), and suppose that CPn E ID2,1 and DCPn --+ 'rf; in L2(W). Take SID2,1 liftings <I1n of CPn. Then for sufficiently small infinite K, <11K is an SL2 lifting of cP and V'<I1K is an SL2 lifting of'rf;. By Theorem 3.7, cP E ID2,1 and Dcp = 'rf; .•

From this it easy to show that D is a local operator; i.e. if cP E ID2,1 then {cp = O} S; {Dcp = O} a.s.

Ocone [13J used the operator D to give a pleasant description of the integrand in the integral representation of cP E ID2,1, and we can give an intuitive proof using our methods.

Theorem 3.11 Let cP E ID2,1. Then the integrand g in the stochastic integral repre-sentation

cp(b)=cpo+ 11g(b,t)dbt

is given by g(b, t) = E(Dcp(b, t)I.1t).

(3.1)

Proof (sketch) Let cp = E~=o CPn = CPo + E~=1 In(/n) be the chaos expansion of cp; taking appropriate liftings we have cp = 0<11 where

K

<11 = CPo + 0 L In (Fn) n=1

and <11 is an SID2,1 lifting of cpo We can write <11 as

<11 = CPo + L G(x, t)Xt tET

where G(x, t) = 2:~=1 In- 1(Fn(·, t)) (with Fn(s, t) = 0 for (s, t) ~ Ll~). Clearly Gis a nonanticipating SL2 lifting of the adapted integrand 9 E L2(W X [0,1]). Now

t<1 _<t<1

158 N. J. Cutland and Siu-Ah Ng

(since Gis nonanticipating, V.G(x, t) = 0 if 5 ~ t). Using the internal isometry property, this means that

so that (using Theorems 3.2 and 3.4) we see that g(., t) E [)2,1 for a.a. t and VG(x, 5, t) is an SL2 lifting of Dg(b, 5, t). From (3.2) it follows that

D.rp(b) = g(b, 5) + /.1 D.g(b, t)dbt

and taking conditional expectations gives the result .•

The following is a strengthening of Theorem 3.7, whose proof we omit:

Theorem 3.12 Supposerp E L1(W) hasSL1 liftingi:P E OL2(n), andVi:P E °L2(nx T) is an SL1 lifting of'lj; E L2(W X [0, I]}. Then rp E [)2,1 and'lj; = Drp.

Using this we can prove the product rule for D.

Theorem 3.13 Suppose that rp, 'lj; E [)2,1 and rpD'lj; + 'lj;Drp E L2(W x [0, I]}. Then rp'lj; E [)2,1 and D( rp'lj;) = rpD'lj; + 'lj;Drp.

Proof Take S[)2,1 monomial liftings i:P and W of rp and 'lj;. Classical calculus gives V(i:PW) = i:PDw + wDi:P E °L2 n S£1. So rp'lj; E L1(W) and rpD'lj; + 'lj;Drp E L2(W X [0, I]} have S£1liftings i:Pw and V(i:PW) and Theorem 3.12 gives the result .

• It is routine to extend the above theory to iterations of D, so that for suitable functionals rp E L2(W) we have

D2rp(b, 5, t) = D(Drp(·, 5))(b, t).

We define [)2,2 = dom(D2) and it can be easily checked that rp E [)2,2 if and only if 'E n(n - 1)\\rpn\\2 < 00 (and this expression gives I\D2rp\\2). Similar remarks hold for higher derivatives Dm. Clearly we have rp E z(n) if and only if rp E [)2,n and Dnrp = OJ and if rp E z(n) then Dn-1rp = Dn-1rpn = in where rpn = In(fn).

The iteration of the local property of D has the following useful consequence.

Theorem 3.14 Let 0 :f:. rp E z(n). Then rp :f:. 0 a.s.

Proof Without loss of generality we may assume that rpn = In(f) :f:. 0 (otherwise rp E z(n-1)). Let A = {rp = O}. Then by the localisation property, Dnrp = 0 on A x [o,l]n. But Dnrp(b, tb'" , tn) = i(tb" . , tn), so if P(A) > 0 then i = 0 a.s., and the same is true for f, contrary to assumption. Hence P(A) = O .•


Directional Derivatives We conclude this section with a brief mention of directional versions of the operator D. Let h E L2[0, 1], so that It = J h belongs to the Cameron-Martin subspace of C. There are several ways to formalise the idea of "differentiation in L2(W) in the direction ii (or h)". One way - which we present using the above framework - is as follows.

Let H be an SL2 lifting of h. A densely defined operator Dh with domain 102,h is defined by:

Definition 3.15 Let rp E L2(W). Then rp E 102,h if and only if rp has an SL2 lifting 41 with 11(\741, H)II < 00, where (G, H) = LtET G(t)H(t)Llt.

If rp E Zn, Dhrp = 0\7 H4I for any S L2 lifting 41, where

\7H W(x) = (\7W(x),H) = E a!(x) H(t). tET Xt

For rp E 102,\ Dhrp = limn .... oo Dhrp(n) = 0\7 H4I(K) for all sufficiently small infinite K.

From this it is clear that 102,1 C 102,h and for rp E 102,1, we have Dhrp = (Drp, h) , with (-,.) here denoting the inner product in L2[0, 1].

There are basic results for Dh that are parallel to Theorems 3.4, 3.7 and 3.12, and Dh is closed, and has a chain rule and product rule like D.

4 The Malliavin Covariance

Suppose that rp E ID2,1. Then the Malliavin covariance a(rp) E L1(W) is given by

Definition 4.1

(4.1)

More generally, for rp = (rpb"" rpn) each rpi E 102,1, a( rp) = (ai,j ( rp)) is a matrix valued r.v. given by ai,j(rp) = (Drpi,Drpj).

One of the chief applications of the Malliavin calculus has been its use in establishing the existence of densities of Brownian functionals, and the following is a simple example of this in our framework (d. Prop. 2.2.1 of [18] and [15] Theorem 3.1).

Theorem 4.2 Suppose that rp E L2(W) with

(i) rp E 102,2 (i.e. rp E 102,1 and Drp E ID2,1)


(ii) 0'( cp) E ID2,1

(iii) O'(cp) -=I- 0 a.s.

Then the measure 11- induced by cp on R is absolutely continuous with respect to Lebesgue measure.

Proof For any real c > 0 let Pe be the measure on n defined by

(4.2)

where we set 0' =O'(Cp), Note that for any 0 ~ n we have lime-+oPe(O) = P(O), and so (considering in particular 0 = cp-l(N) where N is Lebesgue null) it is sufficient to show that l1-e « Lebesgue, where l1-e(') =Pe(cp-l(.)). For this it is well known that we need only show that there is a constant c such that

(4.3)

for any 'IjJ E ci. Take a SID2,1 lifting 4> of cp such that V'24> is SL2 , and a SID2,1 lifting S of 0'. Since 0' ?: 0 a.s. we can take S with S ?: -!c surely. Then (since (V'4>, V'4» is an Stl lifting of 0') we have

J 'IjJ'dl1-e ~ En ('IjJ'(4>(X)) (V';~~4») E((V''IjJ(4», (S + ct1V'4»)

= E('IjJ(4»8((S + ct1V'4»)

< 11'ljJllooE(18((S + c)-lV'4»1

where 8 is the internal classical divergence operator (the dual of V') in *JRT given by

"" "" oU(x, t) 8U(x) = ~ U(x, t)Xt - ~ ox t:.t tET tET t

for U E *L2(n X T) (see the Appendix). The transfer of classical calculus to *JRT

gives

8((S + ct1V'4» = (S + ct18V'4> - (V'((S + ct1), V'4»

= (S + ct18V'4> - (S + ct2(V'S, V'4»

which is SL1 (it is easy to see that 118V'4>1I2 = 11V'4>1I2 + II V'24> 112 - see the Appendix). So c = °E(18((S + ct1V'4>)I) is the required constant in (4.3) .•

We can apply this to show that functionals in the finite chaos' have densities (c.f. [15] Theorem 5.1)


Theorem 4.3 Let ° i: i.p E z(n). Then the measure induced by i.p is absolutely continuous with respect to Lebesgue measure.

Proof Checking the conditions of Theorem 4.2, we have i.p E ID2,2 clearly. The Lemma below shows that O"(i.p) E z(2n-2) and so O"(i.p) E ID2,1; and by Theorem 3.14 we have the condition 4.2(iii), as required .•

Lemma 4.4 If i.p E z(n) then 0"( i.p) E z(2n-2).

Proof Let i.pm = Im(Jm) for m ~ n and take an 8L2 liftings Fm of 1m. So Vel> = ~m<nIm-l(Pm) is an 8L2 lifting of Di.p and 8 = (Vel>, Vel» is an 8Lllifting of 0"( i.p). It can be shown that there is a finite constant c = Cn such that for all kI. k2 < nand Gi : ll{ x T --+ "lR

En ((h 1 (Gd,h2 (G2))4) ~ cIIG l ll4 11G2 11 4

Thus E( 8 4 ) < 00 and so 8 is an 8 L2 lifting of of 0"( i.p) . It is clear that the monomial terms in 8 have degree ~ 2(n - 1) .•

5 The Skorohod Integral

The Skorohod integral au (also called the divergence operator) is defined for certain integrands u E L2(W X [0, I]} and is an extension of the Ito integral that allows u to be nonanticipating. It was first introduced in [16]. The usual standard definitions (see e.g. [10],[11],[14]) of au obscure the fact that it is an integral, whereas the nonstandard approach we outline below makes this quite apparent.

First, for reference, we give the standard definition. A function u E L2(W X [0,1]) has chaos expansion

00

u(b, t) = L un(b, t) = uo(t) + L In(Jn(-, t)) n=l

with In E L2(lln X [0, I]}. The symmetrisation of I = In is the function I E L2(lln+d given by

n+l

l(tb' .. ,tn+d = L l(t1, . .. ,ti-l' ti+l, ... ,ti)' ;=1

Definition 5.1 The Skorohod integral au is the L2(W) sum 00

au = L In+1(Jn) n=O

provided this limit exists.


It is routine to see that a is linear and z(n) C dom( a) for each n. Moreover, au is explicitly given in terms of its chaos expansion, and from this we see that E( au) = O.

For our nonstandard approach, consider first u(b,t) = In(f(·,t)) which has SL2 lifting U(x,t) = In(F(.,t)), where F is an SL2 lifting of f. Defining F in the same way that j was defined, we see that au is SL2 lifted by In+1(F).

Now we have the following crucial Lemma.

Lemma 5.2

( -)() '" () '" 8U(x, t) 2 In+1 F x = L..J U x, t Xt - L..J 8x Xt tET tET t

Proof Routine combinatorics .•

In view of this we make the following

Definition 5.3 Let U E *L2(0 X T). Then Ju E *L2(0) is defined by

FU _ '" U( ) _ '" 8U(x, t) 2 a - L..J x, t Xt L..J 8 Xt

tET tET Xt

Now it is straightforward to prove:

Theorem 5.4 Let u E dom(a) have SL2 lifting U(x, t). Then for all sufficiently small infinite K, JU(K) is an SL2 lifting of au.

Although Ju is a good representation for au for many purposes, a nicer one is obtained by replacing the term x; by D.t. Of course in Anderson's construction of Brownian motion, x; = D.B; = D.t, but this is not true in our model (this is the price to be paid for having Xt = D.Bt vary continuously through "IR, to allow internal differentiation 8/8xt). Nevertheless, if we make the definition

Definition 5.5 For U E *L2(0 X T),

aU = L U(x, t)Xt - L 8~x, t) D.t tET tET Xt

we have the following parallel to Theorem 3.4(a).

Theorem 5.6 Let u E dom( a) with S L2 lifting U. Then for sufficiently small infinite K

E( (JU(K) - aU(K)?) RJ 0

and hence aU(K) is an SL2 lifting of au.


Remark. We have chosen to use the same symbol 8 to denote the standard operator on L2(W x [0,1]) and the internal .operator above on *L2(0 x T); this should not cause confusion since we will use u and U to denote members of the respective domains.

If U is nonanticipating (i.e. U{x, t) depends only on x. for s < t) then 8U{x, t)/8xt = 0. So the usual nonstandard theory of the Ito integral now gives immediately:

Corollary 5.7 If u E L2{W X [0,1]) is adapted, then u E dom{ 8) and 8u = Jo1 udbt (the Ito integral).

The following results parallel Theorems 3.2, 3.4 for D.

Theorem 5.8 Let u E L2{W X [0,1]). Then u E dom(8) if and only if u has an SL2 lifting U with 118UII < 00.

If u E dom{ 8) and U is any S L2 lifting of u, then

(a) 118ull $ °1l8UII (b) 8U is a lifting of 8u if and only if 118ull~ 118UII.

The following parallels Theorem 3.7 and can be applied to show that 8 is closed.

Theorem 5.9 Suppose that u E L2{W X [0, 1]) has SL2 lifting U and 8U is an SL2 lifting of'f/i E L2{W). Then u E dom{ 8) and 8u = 'f/i.

In the literature on the Malliavian calculus, there is much mention of various integration by parts formulae. The most fundamental of these is the following, which shows that 8 and D are dual to one another. It is a simple consequence of classical integration by parts at the nonstandard level.

Theorem 5.10 Let u E dom(8) and cp E ID2,1. Then

E{cp8u) = E(Dcp.u)

i.e. in terms of the inner product in L2{W) and L2{W x [0,1]) respectively

(cp,8u) = (Dcp, u).

Proof Let 4? be an SID2,1 lifting of cp and U be an SL2 lifting of u such that 8U is an SL2 lifting of 8u. Then we have to show that

E{4?8U) = E{V'4?U)


i.e.

This follows immediately from the classical by parts formula for the Gaussian measure on *R (see the Appendix) for each fixed t E T:

E( w(xt)(8(xt)xt - 8~~:t) ~t)) = E(8!~:t) 8(xt)~t).1

There is no standard counterpart of the isometry property for the Ito integral that applies to all u E dom(8). However, if U E *L2(0 X T) then elementary calculations give the following internal isometry:

Theorem 5.11 Let U E *L2(0 X T); then

118UII~ = 11U11~XT + E [E V.U(x,t)VtU(x, S)~t2l .,tET

= 11U112 + ~IIV.U(t) + V t U(s)1I 2 - IIVUI12 < 11U1I2 + IIVUI12

This gives an isometry for the subspace [,2,1 defined by

Definition 5.12 Let u E L2(W X [0,1]). Then u E II},! if u(., t) E [)2,1 for a.a. t and D.u(b, t) E L2(W X [0,1]2).

Hence

Theorem 5.13 If u E [,2,1 then u E dom(8) and

118ul12 IIull2 + E [11 11 D.u(b, t)Dtu(b, S)dsdt]

= IIul12 + ~IID.u(t) + Dtu(s)11 2 -IIDuI12 < lIul12 + II Du ll 2

Proof Simply take a lifting U of u such that V.U(x, t) is SL2 and apply Theorem 5.11.1

There is a natural 'inner product' type formula for E(8u8v) for u, v E [,2,1, obtained from Theorem 5.13 by polarisation, which we leave to the reader to formulate.

Although it is an open question whether 8 is local on dom(8), it is known that 8 is local on [,2,1, and we sketch the proof of this in our framework (cf. [10] Prop. 4.5).


Theorem 5.14 Let u E lL2,l and A ~ C be measurable. If u = 0 a.s. on A x [0,1] then ou = 0 on A a.s.

Proof (Sketch) Take an SL2 lifting of u such that 'V.U(x, t} is an SL2 lifting of D.u(b, t} (and hence oU is an SL2 lifting of ou). Let g(b} = J01 u(b, t}2dt and G(x} = EtETU(x,t}2~t. Pick an infinite M such that for a.a. x,

1 g(Ox} = 0 :::} G(x}::; M

2 g(Ox} # 0 :::} G(x} ~ M

and take an internal smooth nondecreasing function 9:'1R -+ '1R with 9(y} = 0 for y ::; if and y ~ ii, and 0 ::; 9'(y} ::; 2M, so that y9'(y} ::; 4 for all y. Let U(x,t} = 9(G(x))U(x,t}. Then U is an SL2 lifting of u and from the definition of o we have

oU(x} = 9(G(x))oU(x} - H(x}

where H(x} = Et 'Vt9(G(x}}U(x,t}~t = Et9'(G(x}}'VtG(x}U(x,t}~t = 0 a.s.

Now 9( G(x) )oU(x) is an SL2 lifting of l{g;o!o}oU and so (using Theorem 5.9) it suffices to show that H is S L2. We have

IH(x}1 = 9'(G(x))IL(L2U(x,S)'VtU(x,s)~t)U(x,t)~tl t

1

< 29'(G(x}) L U(x, t)2~t( L 'VtU(x, s}2~t2) 2

< 8/1'VU/I

so H is SL2 (since 'VU is) .•

t SIt

6 The Malliavin Operator

In the early development of the Malliavin calculus the Malliavin operator L (defined below) played a prominent role (see [17] for example), but it is now understood that it is not the most fundamental operator in the calculus. The simplest definition is

Definition 6.1 The operator L : L2(W} -+ L2(W} is given by

dom( L) = {ip E ID2,l : Dip E dom( o)}

L= oD

Below we give other characterisations of L beginning with a nonstandard one using the following internal operator on L2(0).


Definition 6.2 Let ~ E ·L2{0}; define .c~ E ·L2{0} by

.c~{x} = L Y't~{x}Xt - L Y'~~~t = 8Y'~

for ~ E dom{Y';} for all t.

It is immediate that for ~ = In(F} we have .c~ = n~, and so from the results on D and 8 in Sections 3 and 5 we have:

Theorem 6.3 L is the number operator on L2(W}; i.e. cP E dom(L} if and only if L n2 11cpnll 2 < 00 and for such cP

Hence dom(L} = ID2,2 = dom(D2}.

The following results are immediate from the corresponding results for D and 8:

Theorem 6.4 Let cP E L2(W}. Then cP E ID2,2 if and only if cP has an SL2 lifting with 1I.c~1I < 00.

If cp E ID2,2 and ~ is any S L2 lifting then

(a) IILcpl1 :::; °1l.c~II· (b) .c{~(K)} = (.c~}(K) is an SL2 lifting of Lcp for all sufficiently small infinite K.

(c) .c~ is an SL2 lifting of Lcp if and only if 1I.c~1I ~ IILcpli.

We also have

Theorem 6.5 If ~ is an SL2 lifting of cp E L2(W} and .c~ is an SL2 lifting of 7jJ E L2(W} then cp E ID2,2 and Lcp = 7jJ.

Many of the properties of L (and .c) follow easily from those of D (resp. Y') and 8 (resp. 8 internal), including the following consequence of the integration by parts result 5.10:

Theorem 6.6 Let cp, 7jJ E ID2,2. Then

E(cpL7jJ} = EcX[O,l](Dcp.D7jJ} = E(7jJLcp)


The paper [12] gives a large number of identities involving D, 8 and £ which can be obtained easily using the nonstandard framework.

The following is a deeper property of L and is needed to prove the product and chain rules. It is a strengthening of Theorem 6.5 (cf. Theorem 3.12 for D).

Theorem 6.7 Suppose that ep, 'l/J E L2(W) have SLl liftings 4>, \If E ·L2(0) with \If = £4>. Then ep E 102,2 and Lep = 'l/J.

The proof of this involves some deeper analysis of £ as a generator of the OrnsteinUhlenbeck process on 'm.T (which has as standard projection the fact that L is the generator of the Ornstein-Uhlenbeck semi group on L2(W) - see [14] for details).

This result gives straightforward proofs of the following (by transfer of classical calculus on "IRT to suitable SL2 liftings):

Theorem 6.8 Suppose that ep, 'l/J, ep'l/J E L2(W) and ep, 'l/J E 102,2 and 8 E L2(W) where

8 = epL'l/J + 'l/JLep - 2(Dep, D'l/J}.

Then ep'l/J E 102,2 and L( ep'l/J) = 8.

Theorem 6.9 Suppose that fEel and ep E 102,2 and O'(ep} = (Dep,Dep) E L2(W). Then f 0 ep E 102,2 and

Applications to Stochastic Differential Equations In applications to existence of densities of the solutions to SDE's it is necessary to find equations for the evolution of functionals such as Dept, Lept and 0'( ept) = (Dept, Dept) when ept is the solution to an SDE. This is achieved quite easily by combining our approach with Keisler's hyperfinite difference approach to SDE's [8]. This will be explained in detail in [7].

7 Appendix - Hermite Polynomials and L2(R, "y)

We give a brief review of the basics of classical calculus on L2(-y} = L2(lR,,} which is transferred to "JRT in the previous sections. Here, is the usual Gaussian measure on lR so that

The Hermite polynomials hn (() (n = 0, 1,2 ... ) are real polynomials with degree ( hn } = n forming an orthonormal basis for L2(-y), obtained from 1, (, e, e, ... by the


Gram-Schmidt procedure, and we have ho(e) = 1, h1(e) = e, h2(e) = ~(e -1), ... . Alternatively they can be characterised by

There are numerous identities concerning the family (hn)n~o; here we need:

Theorem 7.1 (a) E(hn ) = 0 for n ~ 1

(b) h~ = y'nhn - 1

(c) ehn = h~ + Vn+lhn+l' so E(ehn) = E(h~)

For 0 E L2(-y) we have 0 = L::'=o anhn = Ln On , say, where an = (0, hn) in L2(-y); so 110112 = Ln IIOnll2 = Ln lIanl12. The derivative is extended to L2(-y) by

Definition 7.2

n n

The basic integration by parts formula extending Theorem 7.1(c) is

Theorem 7.3 If 0, 'I/J E dom(V') then eO E L2(,) and

The divergence operator 0 and the operator C are defined by

Definition 7.4

CO = oV'O = eO' - 0"

with dom(C) = dom(V'2).

It is easy to check that the following identities hold.

A nonstandard approach to the Malliavin calculus

Theorem 7.5 For all (}, 'Ij; E L2{t) in the appropriate domains

(a) E((}:l/l) = E(o(}.'Ij;) (so 0 is the dual ofV')

(b) o(} = 2:n vnan-lhn where (} = 2:", anhn

(c) lIo(}1I2 = 2:n(n + 1)a~ = 1I(}1I2 + 11V'(}1I2

(d) Chn = nhn and so C(} = 2:n n(}n

(e) IIC(}1I 2 = lIoV'(}1I 2 = 1IV'(}1I 2 + 1IV'2(}1I2

(f) o((}'Ij;) = (}.o'lj; - 'Ij;.V'(}

169

For application to the Malliavin calculus by transfer, we need an n-dimensional scaled version of this. For the scaling take real e > 0 (we will set e = !:l.t after transfer) and put x = y'c(, so x is N(O,e) under I and induces the measure Ie say on JR. The Hermite polynomials are now H n (x) = hn (x / y'c) and we have H~(x) = Vn/eHn-l(X) etc., giving the integration by parts formula

and, defining o(} = x(} - e(}l, we have the duality

Putting C(} = oV'(} = X(}' - e(}" as before gives CHn = nHn.

To extend to n-dimensions, we have the measure 1: on JRn with orthonormal basis for L2{t;-) given by the family (H,,) for a a multi-index. i.e. a = (at, a2, ... , an) with each ai a non-negative integer and H"(Xt,X2,".,X,,,) = II7=lH,,;(xi). This gives derivatives V'i(} = O(}/OXi and divergences Oi(} = Xi(} - eV'i(} for (} E L2{t;-). The divergence 0 that is dual to the gradient V' = (V't, V' 2, ... , V' n) is o(} = 2:i Oi(}i for (} = ((}1, ..• , (}n). Then

E(o(}.'Ij;) = E('E(}iV'i'lj;e) i

Again putting C = oV', the calculus outlined above extends in a natural way to L2{t;-).

For the application to the Malliavin calculus we simply replace the index set {1,2, ... ,n} by the set T.

References

[1] S.Albeverio, J.-E.Fenstad, R.H!1legh-Krohn, and T.Lindstr!1lm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York 1986.


[2] R.M.Anderson, A non-standard representation for Brownian motion and Ito integration, Israel J. Math. 25(1976), 15-46.

[3] N.J.Cutland, Nonstandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), 529-589.

[4] N.J.Cutland, Infinitesimals in action, J. Lond. Math. Soc. 35(1987), 202-216.

[5] N.J.Cutland (ed.), Nonstandard Analysis and its Applications, Cambridge University Press, Cambridge 1988.

[6] N.J.Cutland & S.-A.Ng, On homogeneous chaos, Math. Proc. Camb. Phil. Soc. 110(1991), 353-363.

[7] N.J.Cutland & S.-A. Ng, An infinitesimal approach to the Malliavin calculus, monograph in preparation.

[8] H.J.Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 297(1984).

[9] H.P.McKean, Geometry of differential space, Annals of Prob. 1(1973), 197-206.

[10] D. Nualart & E. Pardoux, Stochastic calculus with anticipating integrands, Probab. Theor. Rel. Fields 78(1988), 535-581.

[11] D. Nualart & M. Zakai, Generalized stochastic integrals and the Malliavin calculus, Probab. Theor. Rel. Fields 73(1986), 255-280.

[12] D. Nualart & M. Zakai, A summary of some identities of the Malliavin calculus, Lecture Notes in Mathematics 1390, Springer-Verlag 1989, 192-196.

[13] D. Ocone, Malliavin's calculus and stochastic integral representation of functionals of diffusion processes, Stochastics 12(1984), 161-185.

[14] D. Ocone, A guide to the stochastic calculus of variations. In Lecture Notes in Mathematics 1316, Springer-Verlag.

[15] I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Mat. Kyoto Univ. 20-2(1980), 263-289.

[16] A.V.Skorohod, On a generalisation of a stochastic integral, Theo,,, Prob. Appl. 20(1975), 219-233.

[17] D.W.Stroock, The Malliavin calculus and its applications. In Stochastic Integrals, Lecture Notes in Mathematics 851, Springer-Verlag 1987, 394-432.

[18] M. Zakai, The Malliavin calculus, Acta Appl. Math. 3(1985), 175-207.

ERGODIC TRANSFORMATIONS IN AST

Martin Kalina Dept. of Mathematics, Slovak Tech. University, Radlinskeho 11

813 68 Bratislava, Slovakia

Abstract

In this article we investigate the ergodicity of set transformations within the Alternative Set Theory. The main result is that each ergodic transformation is homomorhic to the cycle of a suitable length.

Teturo Kamae in his paper [K] has showen that each "classical" ergodic transformation can be viewed at as being a factorization of a cycle. Our intention is to find out wether this is the only way how to get ergodic transformations.

The paper is written in the language of the Alternative Set Theory [V]. Preliminarily, let us list the cructial notions used in this article. First, an indiscernibility equivalence on a set a is an equivalence, which is a 7l'-class (i.e. it is an intersection of an, at most, countable system of sets) and each infinite set contains at least two elements that are in the same equivalence-class. Such elements are also called in

discernible. If ~ is an indiscernibility equivalence on the set a, then for each x E a

the class {y E a; x ~ y} is called the monad of x and denoted by mon(x). For each

subclass of a, X ~ a, the class {y E a; (3x E X)(y ~ x)} is called the figure of X and denoted by Fig(X). So, Fig(X) is just the union of all the monads of x, such that x EX.

Further, let a be any nonempty set, having a elements, a = {Xi};=l' and c: a -+ a be a set transformation. We will say that c is a cycle if C(Xi) = Xi+1 for 1 :::; i < a and c(xa ) = Xl. More precisely, in that case we will call C an a-cycle. We have already mentioned a factorization of a cycle. More generally, we define the factorization of a set transformation, t : a -+ a, by an indiscernibility equivalence

on the set a, ~, by the following - let T : M -+ M be a transformation, M be the

system of all monads of the elements of a, then T is the factorization of t by ~, if for each x E a (possibly except of a class of Il-measure zero, where Il is the Loeb measure) there is an yEa such that t mon(x) ~ mon(y) and T(mon(x)) = mon(y) hold. The transformation t will be called factorizable by ~.

Finally, a class A is said to be Borel, if it is an element of the least O'-algebra, containing all sets. A relation R is said to be Borel, if R is a Borel class.

171

172 M. Kalina

Now, we are going to introduce the following notation - a will be a fixed infinite set and a the number of its elements, p, will denote the measure, induced by the counting measure and normed by a ,i.e. the Loeb measure ([L], see also [K-Z]) , r will always denote a real number (recall that the system of real numbers is the factor space ~I ==). And finally t will always denote a set transformation with dom(t) = a and rng(t) ~ a.

Lemma 1: Let t be any set-transformation such that p,(x) = p,(t-1 x) for each set x ~ a. Then there is a measurable class X ~ a, for which p,(X) #- 0,1 and C 1 X;2 X hold.

Proof: Take a set 0 #- x ~ a and a , E IN and put

X-y = n{u{Cnx;i ~ n ~,- i};i E FN}

It is just a matter of routine to show that x and, can be chosen in such a way that X-y has all the properties required of X. •

It follows that if we are interested in getting, in some sense, ergodic transformations, we have to shrink our a-algebra. The most natural way how to do it within the framework of the Alternative Set Theory, is to take an indiscernibility equivalence and to restrict our attention to its measurable figures, only.

Let g be a fixed indiscernibility equivalence on a. Denote A the a-algebra of all its measurable figures (i.e. X E A iff X is p,-measurable and X = Fig(X) =

= {y; (3x E X)(y g x)}). We will say that the transformation t is measure

preserving with respect to g if each X E A yealds Fig(t-1 X) E A, i.e. if the figure of its pre-image is measurable, and, moreover, p,(X) = p,(Fig(C l X)). A class X ~ a will be called t-invariant if X ~ C 1 X.

Lemma 2: Let the transformation t be measure-preserving with respect to g. Then

t is factorizable by g. Proof: Let t not be factorizable by g. Then there is a class X E A with p,(X) #- 0 such that for each x E X the image t mon( x) has a nonempty intersection with, at least, two different monads. So, we can find two disjoint classes, Y, Z, such that C 1 Y ;2 X and C 1 Z ;2 X, but, since p,(X) #- 0, t is not measure-preserving with

6. respect to = .•

We will call a measure-preserving transformation t ergodic (with respect to g) if all the t-invariant classes X E A yield p,(X) = 0 or p,(X) = 1.

A straightforward consideration gives

Lemma 3: If there exists a measure-preserving (with respect to g) transformation t then either there are n monads (n being finite), each of them of measure lin, or all monads are of measure O.

Obviously the first case leads to the model of a finite space, which we are not interested in, so, in the whole paper we will consider only indiscernibility equivalences, all monads of which are of measure O.

Ergodic transformations in AST 173

The definition of an ergodic transformation says how the t-invariant classes are if they are figures in g. Something more about t-invariant classes is proved in the following

Theorem 1: Let t be ergodic and let X be a t-invariant class with Fig(X) E A. Then p,(Fig(X)) = 0 or p,(Fig(X)) = 1.

Proof: Obviously Fig(X) ~ Fig(rl (X)). Construct the following chain of classes Xo = FigX, and for i E FN X i+1 = Fig(rl (Xi))' So we get Xo ~ Xl ~ ... ~ ~ Xi ~ ... and the class U{ Xii i E FN} is invariant. The transformation t is measure-preserving, hence p,(Xo) = p,(Xi) for each i E FN, therefore p,(Xi \Xo) = O. It follows P,(U{Xiii E FN}) = p,(Xo), and we get p,(Xo) = p,(Fig(X)) = 0 or p,(Fig(X)) = 1. •

Theorem 2: If the transformation t is ergodic then the class of all finite cycles is of measure O.

Proof: Suppose the assertion does not hold. Then there exists an n E FN so that the set Xn of all n-cycles is of a positive measure. Because of Theorem 1,

p,( Fig( Xn)) = 1. Take an x ~ Xn so that p,( Fig( x)) = 2~' Obviously, the set

U7=l ri (x) = y is invariant and there holds 2~ :::; p,(Fig(y)) :::; ~, and this is a

contradiction .•

Theorem 3: If the transformation t is bijective and the class of all finite cycles is of measure 0, then there exists an indiscernibility equivalence ~ which makes t ergodic.

Proof: The class of all finite cycles is a a-class (i.e. the union of an, at most, countable system of sets) and therefore it can be extended to a set of measure 0, which is invariant (see [K-Z]) , hence, without loss of generality, it can be regarded to be empty.

Fix an irrational real number r E [0,1]. The transformation t consists of f3 cycles, each of them of the length Ii, i < {3. Define a map m : a ~ (Q by the following-

fix a point Xi from the i-th cycle and for all If- < Ii put m(r"(xi)) = If-. tSi , where Ii

tSi tSi E IN is chosen in such a way that E r. Now, it is enough to define the

Ii indiscernibility equivalence ~ by

(Vx,y E a)(x ~ y) ¢:} (m(x) == m(y)/mod1),

and we are done .•

Theorem 4: Let t be bijective and ergodic. Then there exists a set z C a of measure 0 so that tl(a\z) and c are isomorphic, where c : a ~ a is an a-cycle. The isomorphism can be done to be Borel.

Proof: The class of all finite cycles is a, hence there exists its superset z, which is still of measure 0 and t-invariant, and therefore there exists a Borel bijection

174 M. Kalina

between a and a \ z (see [K-Z]). So, without loss of generality, we can assume z = 0.

Let t consist of (3 infinite cycles, each of them of the length "Ii. The isomorphism T will be constructed by the following - take the k-th cycle, i.e. the points Xo up to x6+'n-1 (in a suitable ordering), where 8 = ~~':ll "Ii , and put

T(xE ) = C

T(x6+'n-i ) = 8 - i

for

for

C E n{{19;8::; 19::; "Ik - n}; n E FN}

i E FN \ {O},

and similarly the other cycles, and we are done .•

In the remainder of this paper c : b -t b will denote a cycle, where b will be a set, having a suitable number of elements.

Theorem 5: Let there exist Borel classes A, B, both having measure 0, such that tl(a\A) is homomorphic to CI(b\B), where the homomorphism H is a set function, restricted to the Borel class a \ A. And let

(Vx, Y E b \ B)(I H-1(x) 1 / 1 H-1(y) 1= 1)

holcP. Then there exists an indiscernibility equivalence ~, which makes the transformation t ergodic.

The proof of this Theorem is omitted, since the construction of the searched indiscernibility equivalence is similar to that used in the proof of Theorem 3.

Theorem 6: Let the transformation t be ergodic. Then there exist Borel classes A, B, both having measure 0, such that tl(a\A) is homomorphic to CI(b\B). Moreover, the homomorphism H can be done to be a set function, restricted to the Borel class a \ A, and

holds.

Proof: The Transformation t is ergodic, i.e. the class of all finite cycles is of measure o (due to Theorem 2), hence there exists a sequence of sets {SiljEFN\{O} (subsets of a) with the following properties

1 Sj I/o: - Iii s·nCks. J J = 0

j-l

1 UCi Sj 1 /0: - 1 i=O

and these properties imply the following one

I Sj 1 / 1 C k Sj 1 = 1

for 1 ::; k ::; j - 1

for 0 ::; k ::; j - 1.

(0.1)

(0.2)

(0.3)

(0.4)

Now, we can prolong the sequence {Sj}. Since the equivalence = is 7r, there exists a 1/ E IN \ FN such that for Sv Properties (1-4) still hold.

2For any set z I z I denotes the number of its elements.

Ergodic transformations in AST 175

Denote Ii = U{ t--r Sv; 0 ~ "I ~ v}, Property (3) implies /L(Ii) = 1, and define an Hi : Ii ~ IN by Hli(T) = c-r Sv for "I < v. Since there is no guarantee that c v Sv = Sv holds, we have to restrict Hi in the following manner - denote

A = UiEFN(t-i Sv U t-(v-i) sv) and put H = Hi I (ii\A)' Now it is just a matter of routine to show that H has all the properties required .•

Theorems 5 and 6 say that each ergodic transformation is, up to a homomorphism, an infinite cycle with the "almost" uniform probability distribution.

However, there is still one problem open. Namely, Theorem 2, together with the condition (\Ix E a)(/L(t-ix) = 0), give us an obvious necessary condition on t to be ergodic. But it is unknown (at least to the author) wether this condition is also sufficient or wether there exists a set transformation, fulfilling this condition, which, under no indiscernibility equivalence, is ergodic.

References

[K 1 Teturo Kamae: A simple proof of the ergodic theorem using nonstandard analysis, Israel J. Stat. 42, 1982, 284-290

[ K-Z 1 Kalina, M., Zlatos, P.: Borel classes in AST. Measurability, cuts and equivalence, Comment. Math. Univ. Carolinae 30, 1989, 357-372

[L 1 Loeb, P.A.: Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. AMS 211, 1975, 113-122

[V 1 Vopenka, P.: Mathematics in the Alternative Set Theory, Teubner Texte Leipzig 1979.

N ONSTANDRAD CHARACTERIZATION FOR A GENERAL INVARIANCE PRINCIPLE

Dieter Landers Mathematisches Institut der Universitat K51n,

Weyertal 86 - 90 D-50931 KOln, Germany

Lothar Rogge Fachbereich Mathematik, Universitat - GH Duisburg,

D-47048 Duisburg, Germany

Abstract

In this paper it is shown that every invariance principle of probability theory is equivalent to a nonstandard construction of internal S-continuous processes, which all represent - up to an infinitesimal error - the limit process. This can be applied e.g. to obtain Anderson's nonstandard construction of a Brownian motion on a hyperfinite set.

1 Introd uction and Notation

In 1969 Miiller [8J gave the first nonstandard proof of Donsker's invariance principle. In 1976 Anderson [1 J constructed a Brownian motion with nonstandard methods. He defined a hyperfinite random walk whose standard part was a Brownian motion on a Loeb-measure space. From this construction he deduced Donsker's invariance principle.

In 1986 Stoll [9J constructed Levy's Brownian motion with nonstandard methods from which he derived a new invariance principle. In 1989 Mendieta [7J constructed a Brownian bridge with nonstandard methods and deduced an invariance principle for the empirical process.

The purpose of this paper is to show that in all these cases the invariance principle is not only implied but even equivalent to such a nonstandard construction. More generally we prove that this equivalence holds for each weak convergence result for suitable processes.

The general framework for invariance principles of probability theory is the following:

Let (On' An, Pn ) be probability spaces. Let D be a space of functions from a "time" set T C IRk into IRm , endowed with a topology T. The Borel-algebra 8(V) is the

176

Nonstandard characterization for a general in variance principle 177

cr-algebra cr(7), generated by T. Furthermore let Yn : On -t D be An, B(1J)measurable processes. Then invariance principles are assertions of the form

Wn := (Pn)Yn -t W weakly,

where WIB(1J) are special probability measures (= p-measures) and Yn, n E IN are special processes. (Pn)Yn denote the p-measures on B(1J), defined by

and Wn -t W weakly means convergence with respect to the weak topology, i.e.

liminfWn(O) ~ W(O) for all 0 E T. nEIN

In many invariance principles W is the Wiener measure or the distribution of the Brownian bridge and the function space D is the space C[O, 1] of continuous functions on [0,1] or the space D[O,l] of functions on [0,1]' which are right continuous and have left hand limits.

To cover more general situations we consider a rather large class of function spaces, which are suitable for all invariance principles, known to the authors. To this aim let T C Rk and D C (Rm)T and denote by Tx: the topology (on D) of uniform convergence on compact subsets of T. Let furthermore C(T, Rm) be the system of all continuous functions I : T -t Rm.

1.1 Definition: A topological space (D, T) is called a sip (= suitable for the invariance principle) -space, if

(i) D C (Rm)T where a ETC Rk is locally compact and connected;

(ii) 7 is regular and fulfills the first axiom of count ability;

(iii) C:= C(T, Rm) c D and C E B(1J);

(iv) In~1 {:} In~1 (In E D,I E C). T TIC

The notation In ~ I means that the sequence In, n E IN, converges to I with T

respect to the toplogy T.

1.2 Examples: The following spaces (D, 7) are sip-spaces

(i) D:= C(Rk, Rm) with 7:= Tx:;

(ii) D:= C[O, 1] or D[O, 1] or B[O, 1] with 7:= Tx:;

(iii) D:= D[O, 1] with the Skorohod topology T. Here B[O,l] denotes the set of bounded functions I [0,1] -t R; for the Skorohod topology see e.g. [3], pp. 111.

178 D. Landers and L. Rogge

In order to apply nonstandard methods, we work in this paper with a polysaturated model.

If (D, n is a topological space denote by ns(*D) the set of all near-standard points of *D with respect to r. If 9 E ns(*D) and fED we write st(g) = f if g~f, i.e. if 9 is infinitesimally close to f with respect to the topology r. For near-standard points y of "IRm denote the standard part of y by 0y. Let D C (JRm)T with T C JRk • If 9 E *D is a function, such that g(t) is a near-standard point of "IRm for each t E T, then 0g denotes the function from T into JRm, defined by Og(t) := O(g(t)) for t E T. A function 9 E *D is called S-continuous, if or" s ~ t E T:::} g(s) ~ g(t).

If (On' An, Pn) are probability spaces and Yn : On -4 D are An, B(D)-measurable processes for n E IN, we denote for h E *IN - IN by

the nonstandard extension of the sequence

n -4 (On' An, Pn , Yn ), n E IN,

at the point h. Then (Oh, Ah, Ph) is an internal probability space and Yh : Oh -4 *D is an internal function with yh- 1(G) E Ah for G E *B(D).

Let (0, A, Q) be an internal probability space. Put for M cO:

Q(M) .- supeQ(A): M ~ A E A},

Q(M) .- infeQ(A): MeA E A}.

Then L(A, Q) := {M en: Q(M) = Q(M)} is a a-algebra over 0 and QL := Q = Q on L(A, Q) is the Loeb-measure of Q. -

2 The Main Results

The following theorem is the main result of this paper. It gives a nonstandard equivalence to weak convergence of stochastic processes. It can be applied to obtain invariance principles as well as nonstandard constructions for the limit process.

2.1 Theorem: Let (D, T) be a sip-space. For n E IN let (On' An, Pn ) be probability spaces and let

Yn : On -4 D be An,B(D)-measurable with Yn(w)(O) = 0 for all wE On.

Let furthermore WIB(D) be a p-measure with W( C) = 1. Then (i) is equivalent to (ii) :

(i) (Pn)Yn -4 W weakly.


(ii) For each h E *IN - IN there holds:

a) Yh(w) is S-continuous for Pf-a.a. wE Oh;

b) PHw E Oh : °Yh(W) E B} = W(B) for all BE B(D).

Let us remark that the measures (Pn)Yn IB(D) are not necessarily Radon-measures. This excludes the application of classical techniques (as e.g. the application of Prohorov's theorem) for proving the weak convergence (Pn)Yn ~ W in Theorem 2.1.

At first we apply Theorem 2.1 to the case that Yn , n E IN, are normalized sum processes. For this application let D = D[D, 1) be endowed with the topology T = Tx: and WIB(D) be the Wiener measure. Then (D[D, 1], Tx:) is a sip-space (see Example 1.2 (ii)) and W(C) = 1. For x E *lR let [x) = max{n E *7L.: n ~ x}.

2.2 Corollary: Let (On' An, Pn), n E IN, be p-spaces and ein : On ~ lR for i = n

1, ... ,n be random variables with mean zero and D < 'Tn := var (2: ein) < 00. Let i=1

Then Yn : On ~ D[D, 1) are An, B(D)-measurable and (i) is equivalent to (ii):

(ii) For each h E *IN - IN there holds:

[ht]

a) ,},; i~ eih(W) is S-continuous for Pf-a.a. wE Oh;

[ht]

b) B(t,w) := O(,},;.~e'h(W)),t E [D,IJ,w E Oh, is a Brownian motion on

(Oh, L(Ah, Ph), Pt} with Pf-a.a. continuous paths.

Proof: It is well known that Yn are An, B(D)-measurable. By transfer there holds

Hence Corollary 2.2 follows from Theorem 2.1 using W(C) = 1. •

Invariance principles for the sum process (i.e. assertions of the form (i) in Corollary 2.2) are known for many different classes of random variables e'n, i = 1, ... ,n, as e.g. under certain conditions for independent random variables, for mixing random variables or for martingale difference sequences. In each of these


cases Corollary 2.2 ((i) => (ii)) leads by means of hyperfinite sums to a nonstandard construction of a Brownian motion with continuous paths. Let e.g.

On := {-1, l}n, An be the power set of On, Pn ( {w}) := 1/2n

and ein(W) := Wi for wE On.

Then part (i) of Corollary 2.2 is a special case of Donsker's invariance principle and hence for each h E *IN - IN

1 [ht]

B(t,w):= O( fL Lw;),t E [O,l],w E Oh vh i=1

is a Brownian motion on the hyperfinite set Oh with P;-a.a. continuous paths. Hence we obtain Anderson's construction [1] of a Brownian motion and this construction is, as Corollary 2.2 shows, equivalent to the invariance pinciple for the sum process.

Similarly as in Corollary 2.2 we can apply Theorem 2.1 to stochastic processes other than the sum process. Now we give an application to empirical processes and obtain results of Mendieta [7]. To this aim let (0, A, P) be a probability space and ei : 0 -+ [0,1], i E IN, independent random variables which are uniformly distributed over [0,1]. Let

Fn(t,w):= .!.#{i:::; n: ei(W):::; t},t E [0, l],w EO n

be the empirical process and put

Yn(W)(t) := v'n(Fn(t,w) - t).

Let D[O,l] be endowed with the Skorohod topology 7 and let WI on B(V) be the distribution of the Brownian bridge. Then (D[O, 1],7) is a sip-space (see Example 1.2 (iii)) and Wl(C) = 1. Furthermore Yn : 0 -+ D[O,l] are A,B(D)-measurable. Since PYn -+ WI weakly (see e.g. [3], p. 141), we obtain from Theorem 2.1, that for each h E °IN - IN :

is a Brownian bridge on (-n, L(OA, °P), opL) with opL_a.a. continuous paths. By transfer there holds:

Fh(t,w) = ~#{i:::; h: ei(W):::; t},t E 0[0, l],w E-n.

For this application we have used Theorem 2.1 ((i) => (ii)) to obtain a nonstandard representation of a Brownian bridge. On the other hand Theorem 2.1 ((ii) => (i)) can be used to prove the invariance principle for the empirical process from a nonstandard construction of a Brownian bridge.

Nonstandard characterization for a general invariance principle 181

To obtain a nonstandard bridge on a hyperfinite space, we can use Theorem 2.1 in the following way:

Consider once more D[O,IJ with the Skorohod topology. Let nn be the set of all permutations of {I, ... ,n} and put

1 Pn{{W}):= I" for wE nn.

n.

Let Xln, ... ,Xnn be real numbers with

Put

n

LXin = 0, ;=1

n

LX:n = 1, i=1

max IXinl -? 0. l:5i:5n n-+oo

tnt]

Yn{w)(t) := L Xw(i)n; i=1

then (see [3J, p. 209) {Pn)Yn -? WI weakly.

Hence it follows from Theorem 2.1 ({i) =} (ii)) that for each h E *IN -IN

[ht]

Bl{t,W):= O(LXw(i)h),t E [0,1J,w E nh

;=1

is a Brownian bridge on the hyperfinite set nh with Pf:-a.a. continuous paths.

3 Proof of Theorem 2.1 and Auxiliary Lemmata

Using the auxiliary Lemmata 1-4, which will be given later, we prove at first our main theorem.

Proof of Theorem 2.1: We show at first:

(C, T n C) is a Polish space,

WI8{D) is a Radon measure. (3.1) (3.2)

To (I): The space (C, Tic n C) is a Polish space according to Lemma 4. As Tn C and Tic n C fulfill the first axiom of countability, we obtain Tn C = Tic n C (use (ii) and (iv) of Definition 1.1).

To (2): Since C E 8(1)) we obtain:

a{T n C) = 8{D) n C c 8(1)).

As W{C) = 1 and measures on Polish spaces are Radon measures, (2) follows using (1).


(i) '* (ii): Let h E *IN - IN be fixed. As Wn := (Pn)Yn --+ W weakly we get that Wh is infinitesimally close to W with respect to the weak topology. Hence (2) and Lemma 3 imply:

Wf(sC 1(B)) = W(B) for all B E 8(V)

ns(*D) = sC1(D) E L(*8(V), Wh )

Since Wn = (Pn)Yn for all n E IN, transfer implies

(3.3)

(3.4)

(3.5)

According to (4) and (5) we can apply Lemma 2 to (n,A,Q) := (nh, Ah, Ph) and Y := Yh and obtain for all B E 8(V) :

pf((st 0 Yhtl(B)) ((Ph)Yh)L(sC 1(B))

(5) Wf(sCl(B))(~)W(B)

Applying (6) to B = C E 8(V) we have

pf{w E nh : st(Yh(W)) E C} = W(C) = 1.

Hence we obtain according to Lemma 1 that there holds for Pf-a.a. W E nh :

Yh(W) is S-continuous and st(Yh(W)) = OYh(W),

From (8) and (6) we obtain (ii).

(3.6)

(3.7)

(3.8)

(ii) => (i) We have to show that Wh is infinitesimally close to W for each h E *IN - IN. Let h E *IN - IN be fixed; we have to show

W( 0) ::; Wf(*O) for all 0 E T

As Yh(W)(O) = 0 for all wE nh we obtain from assumption (ii)a and Lemma 1 that

st(Yh(W)) = OYh(W) for Pf-a.a. wE nh.

This implies for each 0 E T:

W(O) (ii)b pf{w E nh : st(Yh(w)) E O}

< =

(5)

pf{w E nh : Yh(w) E *O} O(Ph{w E nh : Yh(W) E *O} O(Wh(*O)) = Wf(*O). •

The following Lemma 1 is an adaptation of a well known result to our situation.

Lemma 1: Let (D, T) be a sip-space. Then the following two conditions are equivalent for each 9 E *D :

Nonstandard characterization for a general invariance principle

(i) 9 is S-continuous and g(O) is finite;

(ii) g~f for some f E C.

If g~f E C then f = 0g, i.e. f(t) = °g(t) for t E T.

183

Proof: As 7 and 7/c fulfill the first axiom of countability, the identity on D is 7, 7/ccontinuous and 7/c, 7-continuous in each f E C (use (ii) and (iv) of Definition 1.1). This implies for 9 E *D, f E C:

g~f ¢:} g~f =::} f(t) = °g(t) for t E T.

(i) =::} (ii): We prove at first:

g(t) finite for each t E T.

To this aim put

0 1 := {t E T: g(t) finite},02:= {t E T: g(t) infinite}.

(3.9)

(3.10)

Since 0 E 0 1 by assumption and since T is connected, for (10) it is sufficient to prove that

Ob O2 are open.

Let t E 0 1• Then Ilg(t)11 :::; r for some r E 1R+ and S-continuity of 9 implies:

{s E 'T: s ~ t} C {s E *01 : Ilg(s)11 :::; r + 1}.

Hence 0 1 contains a neighbourhood of t, whence 0 1 is open. The proof that O2 is open runs similarly.

According to (10) Og(t) exists for t E Tj put f(t) := °g(t) for t E T. In the usual way it follows that f E C and g(u) ~ *f(u) for all compact points u, i.e. g~f E C. Hence (ii) follows by (9).

(ii) =} (i): According to (ii) and (9) we have 9 ~ f E C. As T is locally compact this implies

g(u) ~ *f(u) for all u E ns('T). (3.11)

From (11) we obtain that g(O) is finite and f(t) = °g(t) for all t E T. S-continuity of 9 follows directly from (11), using f E C .•

Lemma 2: Let (0, A, Q) be an internal probability space. Let (D, n be a regular topological space. Let Y : 0 -? *D be an internal function with

(0:) Y-l(F) E A for all FE *E(V),

((3) ns(*D) E L(*E(V), Qy).

Then for all B E E(V) holds

184 D. Landers and 1. Rogge

(i) srl(B) E L(*B(V), Qy) and (st 0 Y)-l(B) E L(A, Q),

(ii) Q~(srl(B)) = QL((st 0 Ytl(B)).

Proof: Let B E B(D) be given. As Qyl*B(V) is an internal p-content, Corollary 3 (iv) of [4] implies

srl(B) E L(*B(V), Qy) n ns(*D) (~) L(*B(V), Qy).

Now we prove for each G E L(*B(V), Qy)

Qy(G) = (Qy)L(G) = Qy(G).

As srl(B) E L(*B(V), Qy) we obtain

(Qy)L(sCl(B)) (~) Qy(sCl(B)) = 2((st 0 Y)-l(B))

(Qy)L(sCl(B)) (~) Qy(sCl(B)) = Q((st 0 y)-l(B)),

hence (st 0 Y)-l(B) E L(A, Q) and (ii) holds.

To (12) For F E *B(V) we have

(3.12)

(QL)y(F)(:) 0Q(y-l(F)) = O(Qy(F)) = (Qy)L(F). (3.13)

Let c E R+. Since G E L(*B(V), Qy) there exist Fb F2 C *D with

Fl C G C F2 and Fl , F2 E *B(V), (3.14) (Qy)L(F2) - c/2::::; (Qy)L(G) ::::; (Qy)L(Fl) + c/2. (3.15)

Hence we obtain:

2y(G) - < - = L < Qy(G) (M) Qy(F2)(a)(Q )y(F2)

(U) (Qy)L(F2) (~) (Qy)L(G) + c/2

(~) (Qy)L(Fl) + c (~) (QL)y(FI) + c

Qy(FI) + c ~ Qy(G) + c. -(a)

With c ~ 0 this implies 2y(G) = Qy(G) = (Qy)L(G), i.e. (12) .•

Lemma 3: Let (D,7) be a regular topological space. Let WIB(V) be a Radon measure and QI*B(V) be an internal p-content such that Q is infinitesimally close to W with respect to the weak topology. Then for all BE B(V) :

sCl(B) E L(*B(V),Q) and QL(sCl(B)) = W(B).

Proof: For st-l(B) E L(*B(V), Q) see [4], Corollary 3(iv). For QL(st-l(B)) = W(B) use a slight generalization of Lemma 2.6 in [5] or see Theorem 32.6 of [6] .

• Lemma 4: Let T C Rio be a locally compact subspace. Then C(T, R m ), endowed with the topology of uniform convergence on compact sets, is a Polish space.

Proof: A slight generalization of Theorem 31.6 in [2], p. 245


References

[1] R. M. Anderson, A nonstandard representation for Brownian motion and Ito integration, Israel Journal of Mathematics 25 (1976) 15 - 46.

[2] H. Bauer, Ma

- und Integrationstheorie (de Gruyter Lehrbuch, Berlin-New York 1990).

[3] P. Billingsley, Convergence of Probability Measures (John Wiley & Sons, New York-Toronto 1968).

[4] D. Landers and L. Rogge, Universal Loeb-measurability of sets and of the standard part map with applications, Thansactions of the Amer. Math. Soc. 304 (1987) 229 - 243.

[5] D. Landers and L. Rogge, Nonstandard methods for families of T-smooth probability measures, Proceedings of the Amer. Math. Soc. 103 (1988), 1151 -1156.

[6] D. Landers and L. Rogge, Nichtstandard-Analysis (to appear in Springer Verlag, Berlin-New York 1993).

[7] G. R. Mendieta, Two hyper finite constructions of the Brownian bridge, Stochastic Anal. Appl. 7 (1989) 75 - 88.

[8] D. W. Miiller, Nonstandard proofs of invariance principles in probability theory, in Applications of Model Theory to Algebra, Analysis and Probability, ed. W. A. J. Luxemburg (Holt, Rinehart and Winston, 1969) pp. 186 - 194

[9] A. Stoll, A nonstandard construction of the Levy Brownian motion, Prob. Th. ReI. Fields 71 (1986), 321 - 334.

ANDERSON'S BROWNIAN MOTION AND THE INFINITE DIMENSIONAL ORNSTEIN-UHLENBECK PROCESS

Tom LindstrfZjm t Department of Mathematics, University of Oslo,

Box 1053 N-0316 =s10, Norway

Abstract

Anderson's construction [2] of Brownian motion as the standard part of a random walk with infinitesimal increments is one of the success stories of nonstandard analysis. Almost every subsequenct development in nonstandard probability theory is inspired - directly or indirectly - by Anderson's work. The purpose of this paper is to point out how another extremely important process in stochastic analysis - the infinite dimensional Ornstein-Uhlenbeck process - can be derived very easily from Anderson's construction.

Introd uction

Anderson's construction [2] of Brownian motion as the standard part of a random walk with infinitesimal increments is one of the success stories of nonstandard

analysis. Almost every subsequenct development in nonstandard probability theory is inspired - directly or indirectly - by Anderson's work. The purpose of this paper is to point out how another extremely important process in stochastic analysis - the infinite dimensional Ornstein-Uhlenbeck process - can be derived very easily from Anderson's construction.

To explain the basic idea, let us first recall what Anderson did. Choose an infinitely large integer N E ·IN and let fl.t = l/N. Think of

T = {O, fl.t, 2fl.t, ... , 1 - fl.t}

as a hyperfinite timeline. Let n be the set of all internal functions w : T -+ {1, -1} and denote the internal, uniform probability measure on n by P (Le. P(A) = IAI/lnl

tThis research is supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den Norske Stats Oljeselskap a.s. (STATOIL).

186

Brownian motion and the infinite dimensional Ornstein-Uhlenbeck process 187

for all internal sets A). By L(P) we shall mean the Loeb measure of P. Anderson's hyperfinite random walk B : 0 x T ~ oR is defined by

B(w,t) = Lw(s)v'M. s<t

Anderson showed that for L(P)-a.a. w, the internal function B(w,.) is S-continuous, and hence defines a continuous standard function b(w,·) (= °B(w,·)) from [0,1} to R. Moreover, the standard process b(w, t) is a Brownian motion on (0, L(P)) (see, e.g. [I), [2}, [4}, [5}, [6}, or [10} for the details).

The infinite dimensional Ornstein-Uhlenbeck process u is a stochastic process taking values in the space G([O,I)) of continuous functions. Intuitively, it looks like a continuous, random modification of Brownian paths which keeps the Wiener measure invariant. Using Anderson's construction we can make this intuition rigorous in the following way.

Pick an initial element Wo in 0 (and do it in such a way that B(wo,·) is S-continuous). At time 0, toss an unfair coin for each sET to decide whether you want to reverse the sign of s-th component wo( s) or not; the probability of changing the sign should be At/2 and the probability of keeping it (1 - At/2), The resulting path is WAt.

At time At repeat the procedure; flip independent coins for each s and change the sign of each component with probability At/2 to obtain W2At. Continuing in this way, we get a random sequence of elements wo, WAt, .. . , WnAt, ..• in O. Each of these elements corresponds to an Anderson path B(WnAt, .), and hence we have constructed a randomly moving sequence of such paths. Taking standard parts, we get a randomly moving process of Brownian paths. The purpose of the paper is to show that this random motion is an infinite dimensional Ornstein-Uhlenbeck process. Our tools will mainly be taken from nonstandard martingale theory, and all the results we shall need can be found in Chapter 4 of [I}. We shall use the notation and terminology of that reference, and we shall abuse it with the following shorthand: If w E 0 and B(w,·) is S-continuous and induces a standard function x(.), (= °B(w, .)) in G([O,I)), we shall say that w is nearstandard with standard part x.

This paper is organized as follows: In Section 1 we formalize the construction above and study the anatomy of the nonstandard process. In Section 2 we show that the process is S-continuous, and in Section 3 we give a (well-known) representation in terms of a time changed Brownian sheet. In Section 4 we show that the standard part of our process is an infinite dimensional Ornstein-Uhlenbeck process, and Section 5 contains a brief description of the hyperfinite Ornstein-Uhlenbeck operator and the corresponding semigroup. We end the paper with a few remarks on two other nonstandard approaches to the infinite dimensional Ornstein-Uhlenbeck process.

188 T. Lindstr¢m

1 Construction and basic properties

To formalize the construction sketched above, we define the space B of all internal function

where {-l,lV denotes the space of all internal functions from T to {-1,1}. We shall think of e(t) as the sequence of coin tosses performed at time t, and e( t)( s) = -1 will represent the event of switching the s-th component at time t. Let Q be the internal probability measure on B which makes all the events {e(t)(S)h,.ET independent, and which gives the event e(t)(s) = -1 probability l1t/2 (and, consequently, the event e(t)(s) = 1 probability 1 -l1t/2). Given an initial value wo, we define a process

ewo:BxT--+O

by

i.e., the state changes from t to t + l1t by switching the s-th coordinate if and only if e(t)(s) = -1. When the initial value Wo is of little interest, we shall drop the superscript and write e for ewo . We now define the hyperfinite Ornstein-Uhlenbeck process U (= uwo) by

U(e, t)(.) = B(e(e, t),')

In this section we shall study the basic properties of U as a nonstandard object. We begin with a simple observation.

1.1 Lemma. For all 5, t E T and Wo E n

Proof: Let

Since the probability of switching the s-th coordinate between time t and time t+l1t is l1t/2, we have

PHllt = Pt(1- l1t/2) + (1 - pt)l1t/2

and hence PHt.t = Pt(1- l1t) + l1t/2

Solving this difference equation with initial condition Po = 1, we get

Pt = [(l_l1t)t/t.t + 1]/2 ~ (e-t + 1)/2 •


For each sET, let F. be the internal algebra on S generated by the random variables {((t)(r)lt,r E T,r < s}. Whenever we say that something is a martingale in the s-variable, we shall mean an internal martingale adapted to the filtration {F.}.

1.2 Lemma. Fix t and woo Then

EQ[U((, t)(s + ~t) - U((, t)(s) IF.] = wo(s)~ (1 - ~t)t/tlt

= wo(s)~e-t + o(~)

where o( vz:;:t) denotes a quantity which is infinitesimal compared to vz:;:t .

Proof: With Pt as in the preceding lemma, we have

EQ[U((, t)(s + ~t) - U((, t)(s)IF.] = wo(s)vz:;:t . Pt - wo(s)~ . (1 - Pt) =

wo(s)~ (2pt - 1) = wo(s)~ (1- ~t)t/tlt = wo(s)~ e-t + o(~) •

1.3 Lemma. Fix Wo and t. Then

where K(wo,t) is an S -continuous martingale such that the standard part of K(wo,t) /..; e2t - 1 is a Brownian motion. Moreover, for all s, s' E T

where [K(wo,t)] denotes the quadratic variation of K(wo,t).

Proof: The process defined by

r<s

= (1- ~t)-t/tltU((, t)(s) - U((, O)(s)

is a martingale by Lemma 1.2. Observe that since

we have

where [K(wo,t)] is the quadratic variation of K(wo,t). Thus [K(wo,t)] is S-continuous, and so is K(wo,t) according to Theorem 4.4.16 in [1].

190

Observe next that

EQ(~K(wo,t)(s?I.r.) = [wo(s)~ ((1 - ~ttt//);t - 1Wpt +

+[wo(s)~ ((1- ~ttt//);t + 1)]2(1 - Pt)

= ~t{ [(1 - ~t)-t//);t - 1]2(1 + (1 _ ~t)t//);t)/2 +

+[(1 - ~ttt//);t + 1]2(1 - (1 _ ~t)t//);t)/2}

= ~t((l- ~tt2t//);t - 1) = ~t(e2t - 1) + o(~t)

Hence the compensator process (K(wo,t») satisfies

T. Lindstr¢m

and since (K(wo,t) (s)/ v'e2t - 1) ~ s, we see from (the proof of) Theorem 4.4.18 in [1] that the standard part of K(wo,t) /v'e2t - 1 is a Brownian motion .•

So far we have let s vary for fixed t. If we reverse the situation, we first obtain the following result.

1.4 Lemma. Fix Wo and s. Then

EQ(U(e, t + ~t)(s) - U(e, t)(s)IU(e, t)(s) = x) = -x~t

Proof: Let w = eWo(e, t). Since B(w, s) = U(e, t)(s) = x, we must have

s x I{r E Tlr < s and w(r) = I} = A + r;:-;

ut 2v~t

s x I{r E Tlr < sand w(r) = -I} = A - r;:-;

ut 2v~t

When we move from t to t+~t, the expected number of switches among components belonging to the first set is (s/~t + x/2~) . ~t/2 = s/2 + x~/4, and the expected number of switches among components belonging to the second set· is (s/~t - x/2~) . ~t/2 = s/2 - x..(iS:i/4. Since each switch of the first kind changes U by -2..(iS:i and each switch of the second kind changes it by 2..(iS:i, the expected change is -2~ (s/2 + x..(iS:i/4) + 2~ (s/2 - x..(iS:i/4) = -x~t .•

1.5 Lemma. it Fix Wo and s, and assume that Wo is nearstandard. Then

t-/);t U(e, t)(s) = (1- ~t)t//);t(U(e, O)(s) + L (1- ~tt(r+/);t)//);t~W(e, r))

r=O


where W is an S-continuous martingale whose standard part is ..;2S times a Brownian motion. Moreover, if we let

N(WO'.)(~, t)) = ~)1 - ~tt(r+At)/At~W(~, r), r<t

then

for all t, t' E T.

Proof: By Lemma 1.4, the process

W(~, t) = U(~, t)(s) - U(~, O)(s) + L U(~, r)(s)~t r<t

is a martingale (by which we mean an internal martingale with respect to the obvious filtration 9t). Rewriting this definition as

U(~,t)(s) = U(~,O)(s) - LU(~,r)(s)~t+ W(~,t) r<t

we get a difference equation for U. Solving it, we see that

r<t = (1 - ~t)t/At(U(~, O)(s) + L(1- ~tt(r+At)/At~W(~, r))

r<t

Returning to W, we first observe that since the maximal value of U(~, r)(s) is s/..;z;:t :S 1/..;z;:t, we always have

I~W(~, t)1 ~ I~U(~, t)(s)1 + IU(~, t)(s)~tl :S 2.;t;:i +..(i;i = 3.;t;:i

and hence [W(~, t)] - [W(~, t')] ~ 91t - t'l

for all t, t' E T. By Theorem 4.2.16 and Proposition 4.4.3 in [1], this means that W is S-continuous and S-square integrable. It follows that if U(~,O)(s) is finite, then with probability one, U(~, t)(s) remains finite for all t. From this we see that

and hence

EQ(~W(t)219t) ~ EQ(I~U(t)(s) - U(t)(S)~tI219t) =

(4~t + o(~t))s/2 = 2s~t + o(~t)

(W(t)) :::::l 2st

192 T. Lindstr¢m

almost everywhere. By Theorem 4.4.18 in [1], the standard part of w.j2S is a Brownian motion.

Defining N(wo.s)(~, t) = Lr<t(1- ~t)-(r+At)/LH~W(~, r), we finally observe that

Combining Lemmas 1.3 and 1.5, we get:

1.6 Proposition. For any nearstandard Wo

where M(wo)(~, t, s) is an S -continuous martingale in each of the variables sand t when the other is kept fixed. For each t, the standard part of (~, s) -+ M(wo)(~, t, s) is of the form J e2t - 1 w(~, s) where w is a Brownian motion. For fixed s, the standard part of (~, t) -+ M(wo)(~, t, s) is of the form .j2S J erdb(~, r) where b is a Brownian motion.

Proof: Just observe that in the notation of Lemma 1.3, we have M(wo)(~, t, s) = K(wo.t)(~, s), and in the notation of Lemma 1.5, M(wo)(~, t, s) = N(wo.s)(~, t) .•

Remark. It may be helpful to rephrase the results of this section in standard terms. If we keep t fixed and vary s, then Lemma 1.3 tells us that the standard part u(~, t)(s) of U(~, t)(s) can be expressed as

u(~,t)(s) = e-t(u(~,O)(s) + Je2t -lw(s))

where w(·) is a Brownian motion (depending on t).

If we instead fix s and vary t, then the proof of Lemma 1.5 says that u(~, ·)(s) is a solution of the Langevin equation

du(~, t)(s) = -u(~, t)(s)dt + ..;2s db(t)

where b(·) is a Brownian motion (depending on s), and hence u(~, ·)(s) is a onedimensional Ornstein-Uhlenbeck (velocity) process

(see, e.g., Nelson [9]).

Remark. There is an alternative way of expressing the results above in terms of time-changed Brownian motions (or - more precisely - in terms of a time-changed Brownian sheet), but for technical reasons it will be convenient to postpone that discussion till after we have taken a closer look at continuity properties.


2 Continuity

The results above tell us that for each s, the process (~, t) ~ U(~, t)(s) is Scontinuous for L(Q)-a.a. ~, and similarly for (~,s) ~ U(~,t)(s) when t is kept fixed. What they do not tell us, is how the exceptional sets are related, and thus it could happen that U(~, t)(s) failed to be a.s. jointly continuous in sand t. This would certainly create problems for our intended interpretation of U as a C([O, 1])valued, continuous process, but in this section we shall show that U is, in fact, jointly S-continuous a.e. whenever Wo is nearstandard. Note that by Proposition 1.6 it suffices to prove that M(wo)(~, t, s) is jointly continuous.

As our main tool we shall use the nonstandard version of Kolmogorov's Continuity Theorem (see [1], Proposition 4.8.5). According to this result, it suffices to find constants p, K, c E R+ such that

EQ[IM(~, t, s) - M(~, t', s')/P] ~ KII(s, t) - (s', t')1I 2+c

for all s, s', t, t' E T. As a first step toward this inequality, we have the following observation.

2.1 Lemma. Fix Wo and t. Then for any p > 1

where Cp is a constant only depending on p (and not on Wo and t).

Proof: By the Burkholder-Davis-Gundy inequalities (see, e.g., page 126 of [1])

EdIM(~, t, s) - M(~, t, s')/P) ~

~ Pv12 Ed[M](~, t, s) - [M](~, t, S,))p/2 ~ 2PePpv12Is - S'IP/2

where the last step uses that [M](~, t, s) - [M](~, t, s') :::; 4e2ls-s'l (recall Lemma 1.3 and that M(~, t, s) = K(wo.t)(~, s)) .•

We then do the same with sand t interchanged.

2.2 Lemma. Fix Wo and s. For each p > 1,

where Kp is a constant depending only on p.

Proof: Just as above, but replacing Lemma 1.3 by Lemma 1.5 .•

2.3 Proposition. Assume that Wo is nearstandard. Then for almost all~, the process (s,t) ~ U(~,t)(s) is (jointly) S-continuous.

194 T. Lindstr¢m

Proof: As already observed, it suffices to show that M(€, t, s) is jointly S-continuous. Choose p > 4, and note that by Lemmas 2.1 and 2.2

EQ[lM(~, t, s) - M(~, t', s')IP] ::; ::; 2P(EQ[lM(~, t, s) - M(~, t', s)IP] + EQ[IM(~, t', s) - M(€, t', s')IP]) ::;

::; KII(s, t) - (s', t')IIP/2

for some constant K. The proposition follows from Kolmogorov's theorem .•

We can now define the standard part of U as the process u : S x [0,1] ---* 0([0,1]) given by

u(~,t)(s) = "U(~,t')(s')

where t' ~ t and s' ~ s. It follows immediately from Proposition 2.1 that u is continuous (L( Q)-a.e.).

3 The Brownian sheet representation

If m is the standard part of M, then clearly

u(~, t)(s) = e-t(u(~, O)(s) + m(~, t, s))

If we fix t, then according to Proposition 1.6, the process (~,s) ---* m(~,t,s) is of the form Je2t - 1 wee, s) for some Brownian motion w. Another way of expressing this relationship is to say that the process w' defined implicitly by m(~, t, s) = w'(~,s(e2t -1)) is a Brownian motion. If we instead fix s, we know that (~,t) ---* m(e, t, s) is of the form V2S J erdb(r), which means that it is a continuous martingale with quadratic variation

Hence the process b' defined implicitly by m(e, t, s) = b'(~, s(e2t -1)), is a Brownian motion.

With this observation in mind, it is natural to introduce a new random field v(~, t, s) by

v(~,e2t -l,s) = m(~,t,s)

or, put more explicitly,

v(~, t, s) = m(e,ln(t + 1)/2, s)

3.1 Proposition. v is a Brownian sheet with v(O,s) = v(t,O) = O.


Proof: We have to show that v is a continuous, Gaussian field with covariance

E(v(t, s)v(t', 5')) = min(t, t') . min(5, 5').

It is obvious that v is Gaussian, and the continuity was proved in the previous section. To compute the covariance, we may clearly assume that t ~ t'. There are two cases to consider; 5 ~ 5' and 5 > 5'. Since they are easy and quite similar, we only treat the second one.

Observe that

E(V(t,5)V(t',5')) =

= E([v(t, 5') + (v(t, 5) - v(t, 5')]· [v(t, 5') + (v(t', 5') - v(t, 5')]) = = E(V(t,5')2)

since v is a martingale in each variable, and time evolution in the t- and the 5-direction are independent. But then

E(v(t, 5)V(t', 5')) = E(v(t, 5')2) = E(m(ln(t + 1)/2,5')2) = = t· 5' = min(t, t') . min(5, 5') •

3.2 Corollary. We have

u(~, t)(5) = e-t(u(~, 0)(5) + v(~, e2t - 1,5))

where v is a Brownian sheet with V(0,5) = v(t, 0) = 0.

4 Infinite dimensional Ornstein-Uhlenbeck processes

Infinite dimensional Ornstein-Uhlenbeck processes were introduced by Malliavin [7] and plays a fundamental role in the Malliavin calculus. Put briefly, one might say that these processes and their infinitesimal generator - the Ornstein-Uhlenbeck operator - play the same part in infinite dimensional calculus as Brownian motion and the Laplace operator do in finite dimensions.

There are many ways of describing infinite dimensional Ornstein-Uhlenbeck processes. We could have taken the description in Corollary 3.2 as our definition (see Meyer [8]), but it is more conventional to use a characterization which says that an Ornstein-Uhlenbeck process u is a continuous, strong Markov process with values in C([0,1]) generating the semigroup

(1)

196 T. Lindstr¢m

where W is the Wiener measure on C([O,I]) (see, e.g., Watanabe [11]). We shall show that our process u satisfies these criteria.

4.1 Lemma. Assume thatwo is nearstandard with standardpartx. II I: C([O, 1]) --+ R is square integrable with respect to the Wiener measure W, then

Proof: Observe that if I is bounded and continuous, then

EL(QM(u(WO)(~, t)(.)] = °Eo[" I(U(wo)(~, t)(·)] =

= °EQ["/((I- !1t)t/t.t(B(wo, .) + M(~, t, .))] =

= J I(e-tx(.) + vI - e-2t y(.))dW(y)

by nonstandard measure theory and Propositions 1.6 and 2.3. Using the Monotone Convergence Theorem, the result is easily extended to nonnegative, square integrable functions, and the general case follows by treating positive and negative parts separately. •

4.2 Theorem. Assume Wo is nearstandard with standard part x E C([O, 1]). Then u(wo) is an Ornstein- Uhlenbeck process starting at x.

Proof: We know from Proposition 2.3 that u is continuous, and the lemma takes care of (1). It only remains to check that u is a strong Markov process. Observe that for any two initial conditions Wo and wo, UWo (~, t)( s) - UW~ (~, t)( s) = UWO(~, 0)(8) - uwri(~, 0)(8) for all t and 8. Thus if the initial conditions are infinitely close, one process is just an infinitesimal translation of the other. From this the Markov property follows easily (for instance by an appeal to Theorem 5.4.17 in [1] or Theorem 6.8 in [5], but this is certainly an overkill) .•

5 The hyperfinite Ornstein-Uhlenbeck operator

Having completed our serious work, we may amuse ourselves by taking a look at some nonstandard consequences. Define the hyperfinite Ornstein-Uhlenbeck operator L to be the infinitesimal generator

of U. Observe that L acts on the space L2(fl, Q) of all internal functions F : fl --+ R with the inner product (F, G) = ~F(w)G(w)Q(w). For each subset A of fl, let XA E L2(fl) be the function

V( )_{IIw(s) if Ai0 ""A w - 1 if A = 0


Since the set {XA } is orthonormal and has the right cardinality, it must be a basis for L2(O), and hence any element F E L2(O) can be written uniquely as a sum F = L: F(A)XA where F(A) E*R (this is often called the Walsh expansion of F).

ACfl

5.1 Lemma. For all subsets A of 0

where

Am = L (;) (L\t/2)k-l(1 - L\t/2)m-k k<m k add

When A is finite, AlAI is infinitely close to IAI, and when A is infinite, AlAI is also infinite.

Proof:

= (the probability that an odd number of components in A is switched) x

x 2XA(W)/ L\t = k~1 ('tl) (L\t/2)k(1- L\t/2)IAI-k . 2XA(w)/ L\t

k odd

When A is finite, the sum has only finitely many terms; the first is infinitely close to IAI, the others are infinitesimal. The argument for infinite A is similar .•

We have thus found an orthonormal basis of eigenvectors for L, and we see that the eigenvalues are (infinitely close to) the positive integers. The multiplicity of n is the number of subsets of 0 of cardinality n, that is (I~I). This is in agreement with the standard theory where the spectrum also consists of the positive integers (and where the eigenvectors can be expressed in terms of Hermite polynomials, see, e.g., [11]).

As an immediate corollary we get:

5.2 Corollary. Assume that F E L2(O) has finite norm and Walsh expansion

L: F(A)XA' Then

EQ(F(U(W)(~, t)) >::J L F(A)e-tIAIXA(W) ACfl

198 T. Lindstrf{Jm

6 Two alternative constructions

Let us end the paper with a brief look at two other nonstandard constructions of the infinite dimensional Ornstein-Uhlenbeck process, which may be more convenient for some purposes. The first is a very slight variation on the theme we have already presented. As above we start with an element Wo E n, and then pick at random an element ro E T. Switching the sign of the ro-th component wo(ro), we obtain a new element W2At (we have to move in steps of size 2~t to get the right scaling). Again we choose a random and unif~rmly distributed element rl E T and switch the rl-th component W2At(rl) to get W4At. Continuing in this way, we get a random sequence of Anderson paths, and taking standard parts we get a randomly moving Brownian path. Not surprisingly, this motion is an infinitely dimensional Ornstein-Uhlenbeck process.

This construction has some advantages and some disadvantages compared to the original one. Conceptually, it is slightly simpler and more natural, but technically it is a little less convenient as we lose the Markov property in the s-direction (this will, e.g., make Lemma 1.2 somewhat harder to formulate and prove). On the other hand, the infinitesimal generator turns out to be

which is certainly nicer than the approximative expression in Lemma 5.1. It also turns out that A is more naturally related to the discrete Malliavin operators (see [3]). Let us finally observe that since the alternative construction shows that the Ornstein-Uhlenbeck process is the standard part of a nearest neighbor random walk on the space of all Anderson paths (just as finite dimensional Brownian motion is the standard part of a finite dimensional, nearest neighbor random walk), it adds some extra force to the idea that the Ornstein-Uhlenbeck process is really the infinite dimensional counterpart of finite dimensional Brownian motion.

Let us now turn to the second alternative construction. We still keep the discrete timeline T, but this time the state space n will be the space of all internal maps w : T -+ R, and the internal probability measure P on n will be the one making all increments ~w(t) = w(t + ~t) - w(t) independent with distribution N(O, v'3i). Now let all the increments ~w(t) perform independent, one-dimensional OrnsteinUhlenbeck processes (scaled to keep the initial measure invariant). The standard part of this random motion will be an infinite dimensional Ornstein-Uhlenbeck process.

Again this alternative construction has some advantages and some disadvantages compared to the original one. It is less elementary, but it supports the intuitive idea that an infinite dimensional Ornstein-Uhlenbeck process is one where the infinitesimal increments perform independent, one-dimensional Ornstein-Uhlenbeck processes. It also lends itself more easily to the translation of finite dimensional calculus to an infinite dimensional setting. Finally, if one wants to study flows on Wiener space, the discrete setting is very restrictive and unnatural, while the


continuous model offered by our alternative construction works quite nicely.

In my opinion, the three slightly different constructions above all deserve further study. Although none of them can claim to be the simplest approach to the infinite dimensional Ornstein-Uhlenbeck process (the one suggested by Corollary 3.2 is hard to beat), they all offer additional insight into the nature and structure of one of the most important processes in today's stochastic analysis.

7 References

[1] Albeverio, S., Fenstad, J. E., Hj/legh-Krohn, R., Lindstrj/lm, T.: Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, 1986.

[2] Anderson, R. M.: A Nonstandard Representation of Brownian Motion and Ito-Integration, Israel J. Math. 25 (1976), 15-46.

[3] Holden, H., Lindstrj/lm, T.,0ksendal, B., Ubj/le, J.: Discrete Wick Calculus, to appear in T. Lindstr!6m, B. 0ksendal, S. Ustunel (eds.): Stochastic Analysis and Applications.

[4J Hurd, A. E. & Loeb, P. A.: An Introduction to Nonstandard Real Analysis, Academic Press, 1985.

[5] Keisler, H. J.: An Infinitesimal Approach to Stochastic Analysis, Mem. Amer. Math. Soc. 297 (1984).

[6] Lindstr!6m, T.: An Invitation to Nonstandard Analysis, in N. J. Cutland (ed.): Nonstandard Analysis and its Applications, Cambridge University Press, 1988, 1-105.

[7J Malliavin, P.: Stochastic Calculus of Variation and Hypo-elliptic Operators, in K. Ito (ed.): Proc. Inter. Symp. Stoch. Diff. Equations, Kinokuniya-Wiley, 1978, 195-263.

[8] Meyer, P. A.: Notes sur les Processus d'Ornstein-Uhlenbeck, Seminaire de Prob., XVI, Lect. Notes in Math. Vol. 920, Springer, 1982, 95-133.

[9] Nelson, E.: Dynamical Theories of Brownian Motion, Princeton University Press, 1967.

[10J Stroyan, K. D. & Bayod, J. M.: Foundations of Infinitesimal Stochastic Analysis, North-Holland, 1986.

[l1J Watanabe, S.: Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, Bombay, 1984.

Two APPLICATIONS OF NSA IN THE THEORY OF STOCHASTIC DYNAMICAL SYSTEMS

A. Ponosov t Fakultat fur Mathematik, Ruhr-Universitat

D-44780 Bochum, Germany

Abstract

Two examples of how NSA may be useful in stochastics are considered. In Section 1 a new infinite dimensional multiplicative ergodic theorem for generalized cocycles is obtained; in Section 2 a new method of finding stationary solutions (invariant measures) to nonlinear stochastic parabolic equations is proposed.

1 Multiplicative ergodic theorem for generalized co cycles

1.1 Introduction.

The importance of multiplicative ergodic theorems (MET) is now well-known due to nice applications not only in the theory of dynamical systems (see e.g. [9]), but also in algebra [10]. In this section we propose a new MET for so-called "singular stochastic dynamical systems" which naturally arise from infinite dimensional stochastic equations of various types.

Let (n, F, P) be a (standard) probability space. It is known that the fundamental solution X t ( w), wEn to any linear finite dimensional stochastic differential equation with time-independent coefficients

dXt = AXtdt + BXtdWt

satisfies the so-called "cocycle property":

t,s ~ a

(1.1)

(1.2)

where () : n --+ n is a measure-preserving brownian shift which may be assumed to be ergodic.

tDFG - Research Fellow

200

Theory of stochastic dynamical systems 201

This property allows to define Lyapunov exponents which describe all possible exponential asymptotics of solutions to (1.1):

>.(x):::: lim ~logIIXt(w)xll t-++oo t

a.s.

From Oseledec's multiplicative ergodic theorem it follows that as soon as () is supposed to be ergodic the function >.( x) assumes a finite number of values (not bigger than the order of X t ) which are called Lyapunov exponents corresponding to (1.1) (or to (1.2)) while the set {).(x)} is called the Lyapunov spectrum of (1.1) (or of (1.2)).

This nice picture totally changes if we consider infinite dimensional stochastic equations. Although there exists a number of infinite dimensional generalizations of Oseledec's MET (e.g. Ruelle's MET [14]) providing the existence of a Lyapunov spectrum, they only cover co cycles X t (w) consisting of bounded random linear operators. Such co cycles will be called regular in the sequel. But the thing is that in infinite dimensional spaces there also exist "singular co cycles" which are not covered by any classical MET just due to the fact that they do not consist of bounded random linear operators. The corresponding examples were found by Skorohod [15] and Mohammed [11] (see also [4]). Thus, the simple scalar delay equation

dx = ax(t - h)dt + bx(t - g)dw h,g ~ 0 (1.3)

generates a regular co cycle iff bg = O. Similar difficulties arise in the theory of stochastic partial differential equations (see e.g. [5]).

Our aim is the following: we will generalize the notion of the co cycle to be able to cover the singular case as well; for such generalized cocycles we will formulate a new MET which can be regarded as some kind of extension of Ruelle's MET for compact co cycles in Hilbert spaces.

1.2 Notation and assumptions

Let H be a separable Hilbert space; (0, f, P) be a probability space; (}t : 0 --t o (t E T where T = IN U {O} or lR+ U {O}) be a semi group of measure preserving ergodic maps; fo C f be a a-subalgebra such that ft :::: (}-t fo :J fo; P(A, B) denotes the set of all random points (or more precisely, the set of all P-equivalent classes of random points) in a random subset B = B(w) of H which are measurable w.r.t. a a-algebra A; the set P(A, B) is always supposed to be endowed with the topology of convergence in probability. We also denote by et : P(j, H) --t P(j, H) the isometry given by

Assume now Tt : P(jo, H) -+ P(ft, H) to be linear operators with the following properties:

202 A. Ponosov

AI. Tt are local: x I A = 0 a.s. implies Ttx I A = 0 a.s. and continuous in probability.

A2. Tl is tight: T1(bounded sets) C (tight sets).

A3. The family Tt satisfies the following generalized cocycle property:

(1.4)

A4. sUPllxll$1 E sUPo::;t$1log+ IITtXllH < 00 x E Lco(Jo, H).

Let us make some comments. Property Al trivially holds for bounded random operators, i.e. for the case of Tt given by

(Ttx)(w) = Xt(w)x(w) (1.5)

for some measurable family X t (w) of bounded operators. In this case the generalised cocycle property (1.4) is easily seen to coincide with the ordinary co cycle property (1.2). In finite dimensional spaces there are no linear and continuous (in probability) co cycles different from those given by (1.5). By the way, this roughly explains why singular systems cannot occur in the finite dimensional situation. It is important to observe however that it is not the case any more in infinite dimensional spaces.

On the other hand, it can be easily checked that an autonomous linear stochastic differential equation gives rise to a generalized cocycle if the equation possesses the property of existence and uniqueness of solutions continuously depending (in probability) on initial data. No regularity condition is required. For instance, the delay equation (1.3) gives rise to the generalized cocycle (1.4) for any a, bE IR; g, h ~ o (and to the regular cocYcle only if bg = 0). One may therefore say that it is the property of locality (i.e. AI) which enables us to cover the case of singular systems too.

Assumption A3 extends the notion of a compact cocycle. Assumption A4 corresponds to Oseledec's integrability condition.

1.3 Multiplicative ergodic theorem

Theorem: Under assumptions A1-A4 there exist a decreasing sequence {Am} which goes to -00 or contains it and a flag {Em(w)}, El(W) = H of random subspaces (Oseledec's subspaces) such that

1) Tt(P(Jo, Em)) c e-t(p(Jo, Em)) (invariance of Oseledec's subspaces);

2) tlog II (Ttx)(w) 11-+ Am a.s., t -+ +00 if x E P(Jo,Em\Em+1).

Some comments. One cannot make use here of Kingman's subadditive ergodic theorem here as Ruelle did, because the random values sUPllxll$1 II Tt(w)x II are infinite in the singular case. This is of course the serious obstacle which makes it impossible


not only to use Ruelle's approach, but even to define Lyapunov exponents in the standard way as limits of norms of suitably chosen random operators. By this reason the only thing we can try to make use of is Oseledec's proof which is based on two ideas: a transformation of the cocycle to a triangular form and an extension of the shift. Of course, Oseledec's approach is very much related to the finite dimensional technique, for example operators in Hilbert spaces have no triangular representation, in general. But fortunately these ideas of Oseledec still work if we add some portion of NSA. Thus, it becomes possible to transform operators to the triangular form if we replace our Hilbert space H by a hyperfinite dimensional space F for which H C F C * H.

The idea of extending the shift can also be adapted to our situation, although Oseledec's extensions cannot be used directly. Instead of them we take the socalled "Kamae's universal shift" or the hyperfinite cycle 0 (see [7] or [1] given by

O(n) == n + 1 (modN), o ::; n ::; N, N E *1Noo

which is universal in the sense that any measure-preserving shift () in a separable Borel space is a factor of O. Kamae's shift proved to have other interesting properties. Let us briefly formulate one of them: in the case of finite dimensional cocycle the corresponding Lyapunov exponents can be computed as the shadows of the logarithms of the real nonnegative part of the spectrum (in a suitable internal hyperfinite dimensional space) ofthe operator 8-1'1'1 (where Tn is a properly chosen internal lifting of Tn and (8a)(w) == a(Ow)). Notice that such a property does not hold for an arbitrary internal lifting of the shift ().

1.4 Sketch of the proof

Basically, the proof consists of 3 parts briefly discussed below. For the sake of simplicity we will restrict ourselves to the discrete case, i.e. we will assume that T = 1N U {O}.

i). First, we describe how to extend the given cocycle. For this purpose, let us fix a hyperfinite dimensional space F such that H C F C * H, 7r being an internal orthogonal projection onto F. The shift () == ()1 is replaced by Kamae's shift 0 defined on the probability space fl == {O, 1, ... , N - I} equipped with the Loeb measure P generated by the counting measure. The corresponding completion j of the family of all internal sets will play the role of the extended a-algebra. There also exist internal liftings Tn of the local operators Tn defined on the set p(}o, F) for some fo c f, the Hilbert space F being a factor-space of the set of all finite elements of F w.r.t. the subspace {x E F: Ilxll ~ O}. Moreover, since F is hyperfinite dimensional one may nonstandardly represent the co cycle Tn by an internal family of random hyperfinite dimensional matrices Cn(w), wE fl, namely as follows:

204 A. Ponosov

where x : n -t F are internal and fo-measurable. This representation exists due to the nonstandard transfer principle. Of course, the matrices Cn(w) may have infinitely large norms for particular w, but globally, they still generate the continuous local operators Tn. By making use of this representation we can find an internal unitary matrix U == U(w, K,) depending on wEn and also on an arbitrary internal unitary matrix K, in F with the property

K,-lC(w)U(w,K,) = D(w) (w E n) (1.6)

where D(w) is a lower triangular matrix with nonnegative diagonal elements (the idea of such a transformation in the finite dimensional case was proposed by Oseledec). As the next step, we replace the probability space n by a new one {wo, ... ,wN_dgivenbywo == (w',K,'), Wk == CPWk-l, cp(w,K,) == (Ow,U(w,K,)) where W' E n and an unitary matrix K,' in F are arbitrary, but fixed. Clearly, the projection II(w,l\:) = w induces a bijection between n and the new probability space equipped with the Loeb measure generated by the counting measure, Kamae's cycle on n being transformed to the same cycle Wk -t Wk+I (modN) on the new probability space. So bearing in mind this bijection we may replace n by the probability space just constructed without changing the notation. This new space has at least one important advantage in comparison with the old one. Namely, thanks to the relation (1.6) we may now replace our co cycle by one having a triangular form.

ii). Following our observation in the finite dimensional case we define Lyapunov exponents of the cocycle Tn as the standard parts of the numbers log>. where >. E Sp(e-1fD n {>. 2:: O}, the internal lifting Tn of Tn being given by

Then the invariant subspaces corresponding to the pieces {11 : 1111 < >.} of the spectrum of e-1T1 are checked to be of the form p(]o, E>.) for some measurable family E>.(w) of subspaces of H. According to our definition the Lyapunov spectrum is not changed if we transform the co cycle to the triangular form and moreover the spectrum coincides with the union of Lyapunov exponents computed for all diagonal elements in the triangular representation. By making use of the tightness assumption A2 we can prove that the Lyapunov spectrum of a triangular co cycle is a discrete set, the function>. -t E>. assumes a countable number of values p(]o, Em) and the corresponding Oseledec's subspaces Em for the extended co cycle are of finite codimension.

iii). Finally, we come back to the very co cycle Tn and check that the subspaces P(Jo, H) n p(]o, Em) are of the form P(Jo, Em) for some random subspaces Em of finite codimension in H. Now, it remains to verify the asymptotic property 2 of Theorem 1.3. To do this we first notice that our constructions depend on two infinitely large parameters: dimF and N (the period of Kamae's cycle). Without loss of generality we can assume that dimF = N. Of course, the Lyapunov spectrum might a priori depend on N. In fact, by making use of the Oseledec's MET and


of the already established fact that the invariant subs paces corresponding to each piece {J.l : 1J.l1 ~ A, A E I14} of the spectrum of f)-lTl consist of the random points concentrated in certain finite dimensional random subspaces of F we can prove that the Lyapunov spectrum does not depend on the extension of the cocycle.

From the construction of Tn, it is easily follows that the the dependence Tl == Tl (N) of N is internal and moreover, it also makes sense for all finite N for which the top Lyapunov exponent Atop is easily seen to be expressed as follows:

"In E *INoo

Applying now the permanence principle and omitting some technicalities we can check that the last relation is valid for some infinitely large N. This implies the asymptotical property in Theorem 1.3, at least for the top Lyapunov exponent. Now, replacing the space H by Oseledec's subspaces Em defined as above we may reduce the case of an arbitrary Lyapunov exponent to the case of the top one.

2 Stationary solutions to nonlinear stochastic differential equations in Hilbert spaces

2.1 Introduction

In Section 2 we are going to describe a method of finding stationary solutions for nonlinear infinite dimensional stochastic differential equations indexequation!stochastic differential with a monotone principal part. Our main result (Theorem in 2.5) can be regarded as a generalization of one of Browder's results concerning deterministic elliptic equations.

The usual approach to constructing stationary solutions of infinite dimensional stochastic equations is based on two ideas: first of all, one tries to find stationary solutions to finite-dimensional approximations of the very equation and then to check that weak limit points of the set of distribution-measures corresponding to these solutions yield the stationary solutions (or maybe solutions-measures) to the equation in question.

Let us notice, however, that this approach usually requires the path-continuity of solutions on initial data and by this reason it seems to have only been applied to equations of a rather particular form [16], [3] (e.g. with an additive stochastic noise). In fact, this situation is quite similar to that considered in Section 1 because the property of the path-continuity in the linear case is nothing but the property of regularity. So we may suspect that the reason of forthcoming difficulties is determined by the necessity of replacing regular equations by singular ones.

Additional problems arise if we want to study equations containing monotone-type

206 A. Ponosov

nonlinearities, because of a rather special way of proving convergence of approximations.

To overcome these and other drawbacks it is proposed here (as in Section 1) to involve some NSA in our game, namely, we intend to use the machinery of Loeb probability spaces. The motivation for our choice can be explained as follows. First, Loeb spaces enable us a nice opportunity to avoid both weak solutions and solutionsmeasures to the equation in question and to consider strong solutions only. In fact, this situation is quite similar to one in the theory of weak and strong solutions of stochastic differential equations: once we choose a Loeb probability space we do not have to worry about other probability spaces; Loeb spaces are rich enough to contain all stochastic processes which might be of interest (see e.g. [6]). Second, by making use of Loeb spaces we may forget about weak limits of measures. It is important for our purposes because nonlinearities of monotone type we are dealing with are hardly compatible with this kind of a limit. Finally, the third advantage of our approach is that it does not require any regularity properties of solutions and, in principle, even uniqueness of the solutions to the Cauchy problem is not needed.

2.2 Notation and assumptions

Let V C H c V' be a triple, consisting of a separable Hilbert space H, a reflexive Banach space V and its conjugate V', continuously and densely imbedding each in other, the pairing a· b between V and V' coinciding with the inner product in H iff bE H.

The following equation is the main object of the section 2:

dx = f(x)dt + g(x)dw (2.1)

where w is a scalar (just for the sake of simplicity) Brownian motion, f = a + p, 9 = b + q, a, p : V -+ V', b, q : V -+ H are time-independent coefficients which are assumed to satisfy 5 conditions.

Bl. Monotonicity of the principal part:

2(x - y). (a(x) - a(y)) + Ib(x) - b(y)12 ~ 0,

I . I denoting H - norm.

B2. Semi continuity of the principal part:

Xl • a(x2 + AX3) is continuous in A E 1R ('Ix; E V).

B3. Compactness of the subordinate part:


p: V -+ V', q : V -+ V' are compact continuous operators.

B4. Coerciveness:

2x· f(x) + Ig(xW ~ -fllxllf, + const (f> O,p ~ 2)

B5. Restriction On Growth:

Similar conditions were considered for deterministic case by Browder [2]. We have adapted them for the stochastic case in the spirit of works [12] and [8], where the Cauchy problem for monotone stochastic equations (p = q = 0 in our notation ) was studied. Our main result states that under conditions B1-B5, there is at least one solution which is stationary and stationarily related to the increments of the given Wiener process. This assertion can be viewed as a generalization of Browder's results.

2.3 Nonstandard reformulation of the problem

As in the Section 1 we will fix a hyperfinite dimensional space F such that V c Fc* V.

By making use of hyperfinite dimensional projections onto F which are orthogonal with respect to the inner product in H and the pairing in V x V' respectively, we can find internal liftings A, B, P, Q : *V -+ F of the coefiicients a, b, p, q satisfying hypotheses which are analogous to B1-B5.

The nonstandard version of (2.1) can be written down in the usual way:

(2.2)

where W.,., T E T == {-mLlt, ... , 0, Llt, ... , mLlt}, mLlt = 1 is a suitable internal lifting of Wt defined on a certain hyperfinite probability space (n,:F, P), equipped with an internal measure P, the Loeb extension of which will be denoted by P; Llt is a "sufficiently small" infinitesimal.

It is convenient to represent the condition of stationarity in terms of an internal lifting 01" of the brownian shift ()t, namely, in the following manner:

(2.3)

For our further purposes we want 01" to have two properties: 1) 01"(:Fo) C :Fo where :Fo is an internal subalgebra of :F, playing the role of the a-algebra of" initial events" ,

208 A. Ponosov

and 2) the shift or should preserve the measure P. This can be done if we choose a proper hyperfinite approximation of the Brownian motion Wt. It is interesting to remark that Anderson's random walk with the natural shift does not satisfy these two conditions. Also it is not possible to exploit here Kamae's cycle as a lifting of the Brownian shift. One could think that using the condition (2.3) is not a great trick, because one can easily write down its standard analog by means of the Brownian shift on the canonical Wiener space. Let us however notice that such a standard formulation would not be sufficient for our purposes. As a matter of fact, there is no a priori reason to assume a shift generated by a stationary solution we are looking for to coincide with the Brownian shift (and it is indeed not the case). On the other hand, due to nice features of Loeb probability spaces briefly described above we might expect that the NSA approach would allow use to avoid uncertainty in choosing shifts.

2.4 Solvability of the internal problem (2.2), (2.3)

To solve the equations (2.2) and (2.3) we replace (2.3) by the following condition:

8Xo = X~t (2.4)

where 8a(w) == a(O~tw). We reduce therefore our problem to one of finding periodic (in law) solutions with infinitely small period tl.t. Now, the problem (2.2), (2.3) can easily seen to be rewritten as the following internal equation:

a = ra, (2.5)

with a = Xo, ra = 8-1(a + Aatl.t + Batl.W~t). In order to find a solution to the equation (2.5) we introduce a hyperfinite Banach space L to be a set of :Fo-measurable internal functions from n to F equipped with the norm JETl2. Clearly, the nonlinear internal operator r is *-continuous in L. Now we intend to apply a nonstandard version of the well-known finite dimensional fixed point theorem. Namely, if we manage to prove that for all >. ~ 1 the equation

>.a = ra (2.6)

would have no solutions with sufficiently large but still finite norms, then we could be sure that the equation (2.5) has at least one internal solution with a finite norm. From the coerciveness it follows that

+2E(8aF(8a)tl.t) + 2E(8aG(8a)tl.W) + 2EF(8a)G(8a)tl.W:S


+ consUlt + o(~t) ::; Eo?(l- f~t) + const~t + o(~t) < Eci, (2.7)

if Ea2 > c1const + 1. The inequalities (2.7) contradicts, however, the assumption A ~ 1. This concludes the proof of solvability of the equation (2.5) and, hence, of the problem (2.2), (2.3). We only have to bear it in mind that the solution just obtained is not S-integrable, in general, so we cannot take its standard part in the strong topology of the space L. The only thing we can derive from (2.7) is an estimate E*llaliv < 00.

2.5 Standartisation of the internal solution

Let us introduce another Banach space

£ == Lp(n x [-1,1], L(Fo @ B), L(P x m), V),

where L(·) denotes the Loeb extension of the corresponding internal object, m is the counting .measure on T, B stands for the family of internal subsets of T.

Proposition: If X is an internal solution to the problem (2.2), (2.3) satisfying an equality E*llxollv < 00, then the weak standard part of the process X in the space £ is a stationary process which is also stationarily related to increments of the Wiener process Wt and which satisfies the equation (2.1).

Of course, we will omit here the proof of the proposition (which takes about 5 pages). Let us only make some comments. The essential part of the argument is based on the monotonicity technique [2], [8], [12] arranged in the spirit of NSA. We cannot but remark that in some points our approach still differs very much from those presented in the above papers.

As an immediate corollary we get

Theorem: Under the assumptions Bl - B5 there exists at least one (weak) stationary solution to the equation (2.1) defined on a Loeb probability space which is an extension of the initial probability space. This solution is also stationarily related to increments of the Brownian motion Wt.

Remark: If we want to return to the probability space on which the equation (2.1) is originally defined, we have to take shadows of solutions in the topology of the weak convergence in measure. In this way, we get a "stationary measure" instead of the ordinary process. If now the equation (2.1) has the properties of existence and uniqueness of solutions continuously depending on initial data (i.e. if the equation

210 A. Ponosov

(2.1) gives rise to a "generalized flow of solutions" in the terminology of the previous section) then the "stationary measure" just obtained can be viewed as an invariant measure w.r.t. this generalized flow.

References

[IJ S. Albeverio, J. Fenstad, R. H!1legh-Krohn, T. Lindstr!1lm. Nonstandard methods in stochastic analysis and mathematical physics. Acad. Press. Orlando (1986).

[2] F. E. Browder. Nonlinear elliptic boundary value problems. Bull. AMS, Vol. 69, Nov. (1963), p. 862-874.

[3] A. B. Cruzeiro. Solutions et mesures invariantes pour des equations d'evolution stochastiques du type Navier- Stokes. Expositiones Mathematicae 7 (1989), p. 73-82.

[4] G. Da Prato, J. Zabczyk. Stochastic equations in infinite dimensions. Cambridge Univ. Press (1992).

[5] F. Flandoli. Stochastic flows and Lyapunov exponents for abstract stochastic PDEs of parabolic type. In: Proceedings of Conf. "Lyapunov exponents". Oberwolfach (1990). Eds. L. Arnold et al. Lect. Notes Math. 1486.

[6] D. N. Hoover, E. Perkins. Nonstandard construction of the stochastic integral and applications to stochastic differential, equations. I, II. Trans. Amer. Math. Soc., 275 (1983).

[7] T. Kamae.A simple proof of the ergodic theorem using nonstandard analysis. Israel J. of Math. 42 (1982), p. 284-291.

[8] N. V. Krylov, B. L. Rozovskii. Ito equations in Banach spaces. J. of Soviet Math. Vol. 16, No.4 (1981), p. 1233-1277.

[9J Lyapunov exponents. Eds. L. Arnold et aI., Proceedings, Oberwolfach (1990), Lect. Notes in Math. 1486.

[IOJ G. A. Margulis Discrete subgroups of semisimple Lie groups. Springer (1991 ).

[l1J S.E.A. Mohammed. Stochastic functional differential equations. Pitman, Boston (1984).

[12J E. Pardoux. Sur des equations aux derivees partie lies stochastiques monotones. C. R. Acad. Sci., 275, No.2 (1972), AlOl - Al03.


[13] Ph. Protter. Semimartingales and measure preserving flows. Ann. Inst. Henry Poincare, 22, No.2 (1986), p. 127-147.

[14] D. Ruelle. Characteristic exponents and invariant manifolds in Hilbert space. Ann. of Math. 115 (1982), p. 243-290.

[15] A.V. Skorohod. Random linear operators. Reidel (1984).

[16] M.1. Visik, A. I. Komech, A. V. Fursikov. Some mathematical problems of statistical hydro mechanics. Russian Math. Surveys. V. 34, No.5 (1979), p.149-234.

NONSTANDARD METHODS AND THE SPACE OF EXPERIMENTS

David A. Ross Department of Mathematics,University of Hawaii,

Honolulu, Hawaii 96822, U.S.A.

Abstract

This note gives natural nonstandard proofs of two results that are traditionally considered difficult in the standard theory: the first one is the proof of compactness of the space of experiments in an appropriate topology. The other one is concerned with the connection between two different notions of eqivalence of experiments.

Keywords: nonstandard hull, experiment, statistical experiment

1 Introduction

Following Blackwell, a statistical experiment is a family of probability measures on some fixed measurable space. It is conventional to say that two experiments are equivalent if, for any decision made based on one experiment, one can make at least as good a decision based on the other. The collection of equivalence classes forms a set, which - properly topologized - is called the space of experiments.

This note gives natural nonstandard proofs of two results that are traditionally considered difficult in the standard theory:

(1) In §4, it is proved that the space of experiments is compact when given a certain weak topology. As usual for nonstandard compactness proofs, it suffices to find a standard part for every *experiment. In this case it turns out to be an artifact called the nonstandard hull of the *experiment, obtained by applying the Loeb construction to each of the *experiment's internal probability measures.

(2) In section 5, a different notion of equivalence for experiments is considered; two experiments are equivalent in this sense if each can be used to simulate the other. It is shown that this kind of equivalence implies the other. The proof here is also a kind of nonstandard hull argument: the operator which formally represents the simulation is the standard part of an internal operator.

212

Nonstandard methods and the space of experiments 213

The reader might notice that many standard results are stated without proof, and that some (true) results are omitted where their inclusion would be natural (for example, the converse to Theorem 5.1). This is done to maintain the brevity of this note; a companion article to this one will contain these and related results (see section 6.)

For standard proofs and general overview, the reader is referred to [L] and [8]. The reader is assumed to be familiar with the standard theory, for which [C] is a good reference.

2 Standard concepts

Fix once and for all a parameter set T.

An experiment (on T) is a family E = (X, A, Pt)tET of probability measures on a common measurable space, indexed by T.

Denote by £(T) the class of experiments on T.

If E E £(T) and T' ~ T, denote by E(T') the restriction (X, A, Pt)tETI of E to T'.

If (X, A) is a measurable space, denote by M(X) the set of signed real measures on (X, A), and by M 1(X) the set of probability measures on (X, A).

If (X, A) and (Y, B) are measurable spaces, then a transition kernel from X to Y is a function P : X ---+ M 1(Y) such that for any B E B, the function x I----t PxB is measurable.

More generally, suppose M' ~ M(X), and denote by £b(y) the space of bounded real-valued measurable functions on Y. A generalized transition kernel from M' to Y is a bilinear function P : M' x £b(y) ---+ lR such that (1) if f 2:: 0 and Jl 2:: 0, then p(Jl,!) 2:: 0; (2) Ip(Jl,!)1 :::; IlflIIIJlII; and (3) p(Jl,I) = Jl(X). (In (2), use the supremum norm on £b(y) and the total variation norm on M (X).)

Observe that every transition kernel gives a generalized transition kernel in a natural

way, namely (Jl,!) I----t l [f(Y)dPx(Y)dJl(X)' Moreover, a generalized transition

kernel P induces an operator Mp : M' ---+ M(Y) by Mp(Jl)(A) = P(Jl,XA). (Here XA is the characteristic function of A.)

If E = (X, A, Pth E £(T), then denote by M«E) the set of Jl E M(X) such that whenever 1I E M(X) and 1I 1.. Pt for all t E T, 1I 1.. Jl as well. (Note that for finite T this is just the set of measures dominated by L:T Pt. The general concept is due to Le Cam, and called the L-space of E.)

A decision problem is a family D = (D, Wt)tET where D is a measurable space (note the measure algebra is implicit) and each Wt E £b(D). (Often it is assumed that D is topological or even metric, and that each Wt is continuous; all the arguments below go through with little or no change under these alternative assumptions -indeed, in the standard theory it is proved that without loss of generality D is the

214 D. A. Ross

space consisting of just two points.)

If E = (X, A, Pth and F = (Y, B, Qth E £(T), ( ~ 0, and V is as above, write E;;;)1, F provided for every generalized transition kernel p: M,dF) x £}(D) -+ IR there is a generalized transition kernel",: M<:(E) x .cb(D) -+ IR with ",(Pt, Wt) :::; p(Qt, Wt) + (11Wtll for each t E T. Write E ;;;)f F if E ;;;)1, F for every V; note attention may be restricted to those V where IIWt l1 = 1 for each t. If E ;;;)f F ;;;)f E then E ",f F. The relations ;;;)~, ;;;)0, and ",0 will be written ;;;)1', ;;;), and "', respectively.

Proposition 2.1: £(T)/ '" is a set.

Suppose E = (X, A, PdT E £(T), and 0: ~ T is finite. Put POI = EtEa Pt, and

for tEo: let ::: be the Radon-Nikodym derivative of Pt with respect to POI'

Without loss of generality (:::) a is a measurable function from X to sa =

{x E [0,1] 01 1 Ea Xt = I}. For any /-L E M(X), write p, for the image measure of

/-L on sa under the mapping (:::) a' Note (in particular) thatEa Pt = Pa , and

that /-L t---+ Ii is a linear isometry from M<: (E(o:)) onto M<: (E(o:)), where E(o:)

is the experiment (sa, Borel, Pda.

The following is standard and not difficult.

Lemma 2.2: If E = (X, A, Pt}T and F = (Y, B, Qt)T E £(T), and 0: ~ T is finite,

then E(o:) '" F(o:) iff Pa = Qa. Thus, for any finite 0: the map E(T) t---+ E(o:) induces a map from £(T)/ '" to M(sa); the smallest topology on £(T)j '" making all these maps continuous, where M(sa) is given the weak topology, is called the weak topology on £(T)/ "'. (In practice it will usually be convenient to abuse notation and refer to the weak topology on £(T).)

3 Nonstandard concepts

Assume that the model is saturated at least in the cardinality of £(T).

It will be convenient to adopt the following notation. If (X, A, P) is an internal *probability space, denote the corresponding (standard) Loeb space by (X,.4, P). If (X, A, P,) is a standard probability space, write (X,.4, P) instead of (;X,;::4, ;P).

If E = (X, A, Pt)tE*T E *£(T), let E = (X,.4, Pt}tE*T; this is the nonstandard hull of E. Note that E E £(*T), and so E(T) E £(T). If E = (X, A, Pt)tET E £(T), write E for ;e.


4 Weak compactness

Theorem 4.1: Let E = (X, A, Pt)*T E *£(T); then E(T)/ '" is a weak standard part of Et "'.

Proof: A sub basic open neighborhood of E / '" has the form U = {F = (Y, B, Qth E

£(T)IIJ 'PdPa - J 'PdQal < (.}/ "', where a ~ T is finite, (. > 0, and 'P : sa -+ 1R

I · '1 'fi d h dPt l'f dPt C h h is continuous. t IS easl y ven e t at dP I ts --=- lor eac tEa, so t at a dPa

( dPt). (dA) "'P 0 dP hfts 'P 0 --=- . a a dPa

Morever, 'P(sa) is compact, so this lifting is S-

bounded. It follows: a

(where the two equalities are from a simple change of variables). Thus, Et '" E "U. This proves the theorem .•

Corollary 4.2: £(T)/ '" is weakly compact.

Proof: Every element of "£(T)/ '" is nearstandard by 4.1. •

5 Comparison of Experiments

Theorem 5.1: Let E = (X,A,Pt)T, F = (Y,B,Qth E £(T), and suppose E", F. Then there is a generalized transition kernel TJ : M«:(E) x Cb(y) -+ 1R with MT](Pt ) = Qt, t E T.

Proof: Fix a ~ "T hyperfinite with tEa for every t E T. Fix any 1I E* M«: (E(a)) n "MI(X), e.g. any OPt witht E a. Note "Pa = "Qa on "sa. Define

a map E> : "M«:(E) -+ M«: (E(a)) by E>(Jl) = JI.«: + JlJ.(X)'iI, where (Jl«:,JlJ.) is the Lebesgue decomposition of Jl with respect to "P a = *Qa, and Jl«: « "P a'

Define P : "M«:(E) x" Cb(y) -+ "1R by

p(Jl,J) =" J f(y) [d:t) 0 " (:gJ J d"Qa(Y)

Note if Jl E M«:(E) and f E Cb(y) then

216 D. A. Ross

< 11/111* J d8il-') 0 * (dQt ) d*Q I = 11/1118(* )(* sa)1 dQa dQo. a a I-'

11/11 1*1-'< (X) + ·I-'.l(X)v(X) I 11/11 I*I-'(X)I

< 11/11111-'11 it follows that.,., = 0p is a well-defined function from M«E) x £b(y) to IR with 1""(1-',1)1 ~ 111-'1111/11· Furthermore,

and

""(1-',1) ::::i p(*I-', ·1) = 8(·I-')(*sa)

·I-'«X) + ·I-'.l(X)v(X) = I-'(X),

(where the penultimate equality follows from the observation that d~t is projection dPa

to the tth coordinate.)

Similarly, it is easy to verify that.,., is bilinear and that .,.,(1-', I) ;::: 0 if 1-', I ;::: OJ this proves the theorem .•

6 Discussion

As mentioned in the introduction, some of the results unproved in this note admit interesting nonstandard proofs; these and other results will appear in [R]. For example, E '" F if and only if E( 0.) '" F( 0.) for every finite 0. E T. This in turn implies Proposition 2.1, and also implies that £(T)j '" is weakly Hausdorff (so that in Theorem 4.1, Ej '" is the weak standard part of E).

There are also questions which arise in the nonstandard theory with no standard counterparts; for example:


(I) Is every standard experiment equivalent to its nonstandard hull? (Answer: Yes.)

(II) If two standard experiments are randomization equivalent (i.e., using generalized transition kernels), are their nonstandard hulls randomization equivalent using non-generalized transition kernels? (Answer: Yes.)

(III) If two nonstandard hulls of experiments are randomization equivalent, are they randomization equivalent using non-generalized transition kernels? (Answer: Unknown.)

References

[C] Cutland, Nigel (ed.): Nonstandard Analysis and its Applications, Cambridge University Press, Cambridge, England, 1988

[L] Le Cam, Lucien: Asymptotics in Statistics, Springer-Verlag, New York, 1990

[S] Strasser, Helmut: Mathematical Theory of Statistics, de Gruyter, Berlin, 1985

[R] Ross, David A.: The Nonstandard Hull of an Experiment, in preparation

Part III:

MATHEMATICAL PHYSICS

INFINITE RANGE FORCES AND STRONG L 1-ASYMPTOTICS FOR THE SPACE-HOMOGENEOUS

BOLTZMANN EQUATION

L. Arkeryd Dept. of Mathematics, Chalmers University of Goteborg,

Sven Gultius Gata 6, S-41296 Goteborg, Sweden

Abstract

The usefulness of NSA in kinetic theory both as a technical and conceptual tool is by now well documented. In the present paper we summarize the picture, concerning NSA methods to prove strong Li-convergence with time to Maxwellian limits for standard solutions of the Boltzmann equation starting far from equilibrium. As a new result we prove the strong Li convergence for the space homogeneous BE without angular cutoff.

1 Introd uction

This paper discusses the Boltzmann equation without exterior forces

(at + vax)! = Q(f, f)

with collision operator

The discussion makes sense in any ]Rd, d;::: 2, but we stick to the case d = 3 for convenience. Then the space and velocity arguments of !{, !~, ft, h are (x, vD, (x, v~), (x, Vi), (x, V2) with v~ = Vi + nlvt - v21 cos B, v~ = V2 - nlvi - v21 cos B, and n is a unit vector in S2; in angular coordinates

{( B, 0; 0 S B S 7r /2, 0 S ( S 27r}, dn = sin BdBd(.

The weight function B is

221

222 1. Arkeryd

«(3 = (k - 5)/(k - 1) for inverse k:th power forces) and 0 < b(fJ), a measurable function on (0, 7r /2).

Suppose in the nonstandard context that

If f is nonnegative and 8-integrable on * R3, then Loeb a.e. on ns*(IEe x 1R? x 8 2 )

(2)

This relation provides useful information about f.

Proposition 1 Let q E *L~ (JR3) be given with

J q(v)(1 + v 2 )*dv, J qlvllogq(v)*dv

finite, and with q satisfying (2) for Loeb a. e. (Vb V2, n) E "(JR3 X JR3 X 82). Then either q R:;j 0 for Loeb a.e. v E ns*JR3 or °q(v) > 0 for Loeb a.e. V E ns*JR3.

For a proof see [3J. Having this result, the well-known proof that equality in (2) implies q Maxwellian, can easily be adapted to prove

Proposition 2 Under the hypothesis of Proposition 1, there are a, b E 1E4, and c E JR3, such that for Loeb a.e. v E ns*JR3

q(v) R:;j aexp( -b(v - c?).

For a detailed proof, see [2J. We next discuss the (standard) background of the assumption (1). Start from nonnegative functions f, fn(n E IN) in Ll(JR3) with

wlim meaning weak limit in £1(B). x B). x 8 2 ) for A> 0,

Essentially by a convexity argument it holds for (fn)'{' and f that

Proposition 3 [6J If b is integrable then

where e is defined in (1).

Infinite range forces for the Boltzmann equation 223

In particular if lime(fn) = 0, then it follows that

fU~ = hh a.e. in IR? x IR? x S2,

and so (e.g. employing Proposition 2) the standard function f is a Maxwellian,

f(v) = aexp(-b(v - c?) a.e. in IR?

This can be used to prove weak L1 convergence to a Maxwellian when time tends to infinity for solutions to the Boltzmann equation in various initial(-boundary) value problems. In this paper the corresponding strong L1 convergence by NSA techniques will be discussed. We shall consider in detail a new strong L1 convergence result in the space homogeneous case without angular cutoff for inverse k:th power forces, k > 3, and later comment on space dependent (as well as space homogeneous) cases with angular cutoff.

2 Strong L1 convergence to equilibrium

Consider the space homogeneous, nonlinear Boltzmann equation in the absence of exterior forces

Dd(t,v) = Q(f,f)(t,v), t > 0, v E JR3

with Cauchy initial data

f(O,v) = fo(v) ~ 0, v E JR3.

For inverse k:th power forces (k > 1), bE Ltoc([O, 71"/2)] but

b(O) '" 171"/2 _ 01(k+1)/(k-1)

for 0 near 71" /2. In this case the integral

r/2

io b( O)dO

does not converge. As a consequence, the two terms in the collision operator Q do not converge separately. But we can use a suitable version of the weak form of the collision term;

{ cp(t,v)f(t,v)dv = iJR3

where

224 L. Arkeryd

For k > 3, f E L""(JR+i Ll(JR3)), the collision integral converges. The presently known solutions of (3) for k > 3 are weak Ll limits of solutions fn with b( 0) truncated near 0 = 7r /2, say bn (0) = b( 0) A n, when

J foe v )(1 + v2 + I log fol)dv < 00. (4)

The integral (4) of fn(t) is then bounded, independently of nand t, together with

1'''0 e(fn(t))dt < C < 00, n E IN. (5)

The weak Ll compactness ensuing from (4) can be used to construct ([1]) solutions f = w limfnf, where (fn') is a converging subsequence of (fn)f. Moreover, J satisfies (4) with a uniform bound for t E ~ and using Proposition 3, also (5). For such solutions f, strong Ll convergence to Maxwellians holds, when time tends to infinity.

Theorem 4 Given any sequence (tk)IN, tk /,00, there is a subsequence (tkf), and a global Maxwellian

M(v) = aexp(-b(v - C)2),

such that for T > 0, strongly in £1([0, T] x JR3),

Proof It follows from the uniformly in t bound on (4) for f(t) that, given any sequence (tk) of times increasing to infinity, and from the ensuing weak Ll compactness of (f(. + tk)heIN' that there is a subsequence (tkf) such that f(. + tkf) -" g(.), weakly in Ll(JR3 x [0, T]) for T > O. The proof will demonstrate that 9 equals a global Maxwellian, and discuss the strong Ll convergence. The first part of the proof is to show that 9 equals a time dependent local Maxwellian

M(v, t) = aCt) exp( -b(t)(v - c(t))2).

For this, notice that there is a countable sequence </>11 </>2, ... offunctions with bounded support in lR3 x ~, such that 9 = M in Ll, if

J g</>jdvdt = J M</>jdvdt, j E N.

Also let the sequence contain Xvp, vXvp, V2Xvp,(II,p E N) where Xvp(v,t) = 1 for v2 :::; 112, t:::; p, Xvp(v,t) = 0 otherwise. Set Mk = {</>l1 ... ,</>k}'

The subsequence (tkf) can be so chosen that f(t + tkf) converges weakly in Ll(JR3) for rational t > O. Given cp E Cl,,,,, , (3) implies that the family JlR3 f(t + tkf)cpdv is uniformly equicontinuous in t, and so J(t + tkf) -" get) weakly in £1(JR3) for all t> O. Using (3) for f(t+tkf) it follows that 9 satisfies (3) for t > 0, and J g(t)cpdv is


uniformly continuous in t. Moreover, for some subsequence of (tkl), which will from here on be denoted (tk),

lk dsl /IRJ (. + tk)</> - g(.)</>dvl < 11k,</> E Mk •

We also assume that (tk) was so chosen that

l.tk+k ds /((1112)' - (l1h))log((fd2),I(flh))Bdvldv2dn < 11k,

which is possible by (5).

(6)

(7)

A large part of the proof from here on relies on NSA. By transfer, in the nonstandard context, (6) and (7) hold for all k E *N. Given K E *]No,,, (6,7) implies in particular for k = K that

{"dsl { (f(t,,+.)*</>-*g*</»dvl~O, </>EM", 10 lIR3 (6')

(7')

From (7') it follows that the integrand in the left hand side is infinitesimal; for Loeb a.e. t E ns*IR+

f(Vb t + t,,)f(V2, t + tIt) ~ f(v~, t + t,,)f(v~, t + tIt) (8)

for Loeb a.e. (Vb V2, n) E ns*IR3 x IR3 X 52. Using Proposition 1-2, we may conclude from (8) that for Loeb a.e. t E ns*IR+

f(v, t + tIt) ~ M(v, t) for Loeb a.e. V E ns*IR3,

with M a local Maxwellian,

M( v, t) = a(t) exp( -b(t)( V - c( t))2).

This result together with (6') gives for </> E UkEN Mk ( eM,,) and T E IR+

o ° { *dtll J(v,t+t,,)*</>*dv- * { 3g</>dvl = l[o,T] *IR lIR

J Ldtll 3 °f(v, t + t,,)o*</>Ldv - 0* ( 3 g</>dvl = (9) ·~n _IR k

= J Ldtl J °M(v, tt*</>Ldv - 0* J g</>dvl·

Taking </> = XlIpl, XlIpV, XlIPv2 and letting v tend to infinity, we get

a(t) o*a(t), b(t) =0* b(t), c = o*c(t) Loeb a.e. t E ns*~,

226 1. Arkeryd

where a, b, c are the Lebesgue measurable functions on IRt. It follows that oM = o*M with M = aexp(-b(v - c?), i.e., for Loeb a.e. t E ns·IRt, M is infinitesimally close to a standard local Maxwellian.

In particular, for ¢> E UkENMk

I { (g¢> - M¢»dvdtl lIR3 X[O.T]

= I {3 (O*gO.¢> - o·Mo·¢»Ldvdtl = In •• IR X·[O.T]

= I! Ldt { Ldv(O*gO.¢> _0 MO*¢» I ~ • [o.T] 1 no· R3

<! Ldtl ( 3 Ldv(O.gO.¢> - °MO*¢>I = 0, • [O.T] 1 no· IR

where the last equality is (9). It follows that 9 = M, i.e. 9 is a timedependent Maxwellian.

We have thus proved

f(v, tt< + t) Rl M(v, t) Rl M(v, t) = g(v, t)

for oeb a.e. (v, t) E ns·IR3 x ns*IRt. This implies for T E ns*IRt, that

( 3 ·x ... l* f(t + tt<) - ·MI*dvdt Rl O. lIR x[o.T]

From here it follows in the standard context that

lim ( a If(t + tk) - Mldvdt =' 0, T E IRt. k-+oo lIR x[O.T]

Finally, for f = M the collision term vanishes in (3), so M is independent of t .•

Remark

1) No standard proof of this standard result is presently known.

2) If besides (4), Ivl'fo E Ll(IR3 ) for some s > 2, then in the case of hard and Maxwellian forces, i.e. k ~ 5, the s-moment is globally in time bounded in Ll, [7). It follows that the energy is conserved, which makes the limit M unique, hence f(t,·) -+ M(·), strongly in Ll(IR3 ) as t -+ 00.

3) The above result holds in particular in the case of angular cutoff, where the first proof was the nonstandard one in [2). In this case a standard proof [8) has later followed.

Using the averaging technique instead of the above equicontinuity arguments, an analogous NSA proof of strong £1 convergence can be obtained in various space dependent cases with angular cutoff. Start, e.g., from the setup in the introduction


of this paper for inverse k:th power forces with k > 2, angular cutoff, and the space variable x in a torus 1R? / Z3 (thus avoiding boundary problems). Consider a DiPerna Lions solution f of the problem with initial value fo satisfying

J fo(1 + v 2 + I log fol)dxdv < 00. (4')

Theorem 5 (4) Given any sequence (tk), tk /' 00, there is a subsequence (tkl) and a global Maxwellian

M(v) = aexp( -b(v - c?)

such that for T > 0, f(· + tk') -t M

strongly in L1(IR? /Z3 x 1R? x [0, Tj), and for t > 0,

f(',t+tk') -t M

An analogous result holds under specular or reverse reflexion for a large class of boundaries.

References

[1) Arkeryd, L., Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rat. Mechs. Anal. 77 (1981), 11-21.

(2) Arkeryd, L., On the Boltzmann equation in unbounded space far from equilibrium, Comm. Math. Phys. 105 (1986), 205-219.

[3] Arkeryd, L., The nonlinear Boltzmann equation far from equilibrium, in Nonstandard Analysis and its applications, Cambro UP 1988.

[4] Arkeryd, L., On the strong L1 trend to equilibrium for the Boltzmann equation, Stud. Appl. Math. 87 (1992), 283-288.

[5] DiPerna, R.J., Lions, P.L., On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math. 130 (1989), 321-366.

(6) DiPerna, R. J., Lions, P. L., Global solutions of Boltzmann's equation and the entropy inequality, Arch. Rat. Mechs. Anal. 114 (1991), 47-55.

(7) Elmroth, T., Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range, Arch. Rat. Mechs. Anal. 82 (1983), 1-12.

(8) Gustafsson, T., Global £P properties for the spatially homogeneous Boltzmann eqution, Arch. Rat. Mechs. Anal. 103 (1988), 1-38.

A NONSTANDARD ANALYSIS ApPROACH TO THE THEORY OF QUANTUM MEAN FIELD SYSTEMS t

Manfred P. H. Wolff Mathematisches Institut, Universitat Tiibingen

Auf der Morgenstelle 10, D-72076 Tiibingen

Abstract

In this article we develop an approach to the theory of meanfield systems in which the relevant algebras of observables are quotients of subalgebras of a hyperfinite tensor product of the one-particle algebra under consideration. So this aproach corresponds exactly to the intuitive picture of meanfield systems as many-particle systems, in particular it underlines the statistical character of meanfield phenomena. As practical applications we bring the treatment of general meanfield dynamics in this paper whereas the fluctuation phenomena are dealt with in [WH] on the basis of this approach.

Introd uction

About 20 years ago K. Hepp and E.H. Lieb laid the foundations for a mathematically rigorous treatment of the dynamics of meanfield systems, including already fluctuation phenomena (see [HL] ). However the algebras of intensive observables and of fluctuation observables were not constructed explicitly in that paper. The

problem concerning the intensive observables was attacked later by different research groups as we will quickly review now:

To this end, let A be a unital C·-algebra, the one - particle algebra, and let An denote the n-fold injective tensorproduct A®n (cf [TA], p. 207 and [BRI], p. 144 where it is called the C·-tensorproduct). Consider the canonical embeddings jn+p,n : An -+ An+p (a -+ a Q9 I p, where Ip = 1 Q9 ••• Q9 1 p times). Then the C*inductive limit l~(An,jn+p,n) =: Aoo is called the quasilocal algebra of the system.

The group around A. Rieckers (see e.g. [DR], [RI] , [UN]) as well as P. Bona [BOI] and others extended this algebra, roughly speaking, by selecting first of all the folium F in the state space S(Aoo) generated by the permutation invariant states and then taking the quotient A:/ J of the second dual A: by the annihilator

tThis reseach is part of the project Operator Algebraic Quantum Optics supported by the Deutsche Forschungsgemeinschaft

228

Quantum meanfield systems 229

ideal J of F. In this way the intensiveobservables appear as limits (in an appropriate weak topology) of means of observables at different sites of the lattice IN, say. This approach was generalized further by Morchio and Strocchi [MS]. Another construction of an appropriate extension of AX) was given in [RW] and was used to develop a general theory of meanfield dynamics in [DWI], [DW2]. In [RW] G. Raggio and R.F. Werner used a certain kind of discrete approximation. More precisely they introduced the space of approximate symmetric sequences. This approach was generalized to the space of approximate symmetric sequences with respect to so-called tagged nets. This generalization appeared necessary in order to consider local and intensive observables at the same time, an effort which is not necessary within the first - mentioned approach.

In this paper we want to propose a third approach which seems to us much easier than these other two - taken knowledge of nonstandard analysis for granted. Moreover our approach allows also to deal with fluctuation phenomena which is not possible in the second approach and which seems to be difficult in the first one. Finally our approach seems to be well-suited to discuss the connection between classical and quantum mechanics, as it is already indicated in [Bol], [DW2], and worked out in [B02], cf. also [DW2].

Even if A is commutative the results on the meanfield dynamics are of some interest: if the local dynamics is dissipative the corresponding dynamics on the algebra of all intensive observables is still deterministic (see 4.8), (2)). This effect was already noticed by Th. Unnerstall [UN] without any reference to the classical situation. The close connection between classical and quantum probability theory appears quite natural within our framework: if the one-particle algebra A is commutative, fluctuations show classical phenomena, if it is noncommutative and simple then fluctuations show also quantum mechanical phenomena (see [WH]).

The paper is organized as follows: the first section contains basic notions and results, the second one is devoted to the reconstruction of the algebra of all local and intensive observables. The third section contains basic definitions concerning meanfield dynamics, whereas in the fourth section we treat an important special case: the polynomial bounded generators. The final section is devoted to some generalizations. There we discuss also the connection between quantum mechanics and classical mechanics. The treatment of fluctuations is to be found in [WH].

Acknowledgements: I would like to thank Prof. Albeverio, Bochum for his encouragement, Prof. A. Rieckers, Tiibingen, from whom I learned meanfield theory, for many valuable suggestions, and Prof. W. Henson, Urbana, and Dipl.-Phys. Th. Gerisch for carefully reading the manuscript.

1 Basic notions and results

We adhere to the notions and notations of the introduction.

230 M. Wolff

1.1) Let V = V(X) be the superstructure over an arbitrary set X such that the one-particle algebra A as well as ~ are elements of some v,.(X). (For notions from nonstandard analysis not mentioned here we refer to [AFHL], [SL], [HULO], [HM]). Viewing the n-fold algebraic tensor product A" of A as the subspace of all

n-linear mappings on A,IN (A' : the dual of A) depending only on the first n coordinates we can easily see that all A" as well as their completions An (with respect to the injective tensor norm) are elements of some V.(X) for a fixed s > r.

1.2) In the following let *V be a polysaturated enlargement of V and let N be an arbitrary infinitely large positive integer. Then A®N =: AN is a well-defined element in *v.(X). As usual we build AN, fin = {a E AN : Iiall is limited }, AN,o = {a E AN : Iiall ~ O}, and AN = AN,fin/AN,O. Equipped with the norm lIall = °llall (" 0" : the standard part map in *~fi,,)AN is a unital C"-algebra.3 We denote the

quotient mapping AN,fin 3 a ---+ a E AN by Q.

1.3) As in the introduction, we consider the embeddings jr,,, : An ---+ Ar (r> n). In this way we may identify" An for n standard with jN,n(* An) whenever this is convenient, e.g. in order to define the algebra A(o) of local observables as A(o) = UnEIN jN,n(" An), where as usual "E = {a E "E: a = "b for some bEE}. Then

I n := QjN,nl"An embeds An into AN and satisfies Jr = I n ojn,r (r:5 n).

Then QIA(o) maps A(o) isometrically onto Un J,,(" An), so we may identify this space with A(o). Its closure is obviously isomorphic to Aoo. Aoo is the infinite CO-algebraic tensorproduct in the sense of [BR1], p. 144.

2 The algebra of intensive observables

2.1) Let S(N) be the group of (internal) permutations of {I, ... , N}. Then for each 1T E S(N) by U,,(al @ ••• @ aN) = a,,-l(l) @ • @a,,-l(N) there is uniquely determined a CO-automorphism on AN (use the transfer principle). Moreover 1T ---+ U" is a representation of S(N). So P = ill E"ES(N) U" is a conditional expectation of AN onto the (internal) CO-algebra of all fixed points of this representation.

Externally S(N) is very large. Nevertheless 1T ---+ U" is a representation and P is a conditional expectation onto the algebra of all fixed points of this representation. (Here the operator T is defined by T(x) = (Tx').) 2.2) Proposition: P has the following properties. (a) For x E "Afi" let x(k) = 1k-l @x @ IN-k. Then

1 N P(jN,l(X» = P(x(l» = N Lx(k) =: x.

k=l

x is limited.

3This is obvious though we could not find an explicit reference for it.


(b) P(jN,n( Xl ® ... ® Xn)) ~ Xl .•• Xn for all standard n and all Xb ••• , Xn E * A fin (C) [x,y] ~ 0 for all x,y E *Afin. Here [a,b] = ab- ba denotes the commutator of a and b. (d) [X,jN,n(a)] ~ 0 for all a E * An,fin (where n is standard) and for all X E * Ajin'

Proof: (a) is obvious. ~ N ~

(b) For k = (kl, ... kn) E {1, ... ,n} set x(k) = xl(kd· ,xn(kn). Moreover for j = 2, ... , n let L:(j) denote the sum over all such k for which exactly j components are equal. Then

1" ~ 1 ~ ~ Xl' 'Xn = Nn L..J(O)x(k) + Nn L..J(j)x(k).

j=2

Now for each j 2 2 the sum L:(j) has exactly r(j) := N· (j) . (n - j)!(~~:) summands of norm less than b = max{j}(llxjll + l))n which is limited by hypotheses.

Since :t# ~ 0 for j 2 2 we obtain

1" ~ 1 ,,~1 n-l Xl"Xn ~ Nn L..J(O)x(k) = N(N _ 1) . . (N _ n + 1)' L..J(O)x(k)(I- N),,(I-r:r-)'

But 1" ~ (N - n)' " ~ P(jN,n(Xl ® ... ® xn)) = N! L..J(x(k)) = N! . L..J(O)x(k);

" clearly this element is limited. So P(jN,n(Xl ® ... ® xn)) - Xl'" Xn ~ (1- 117=1(1- -Iv)). P(jN,n(Xl ® ... ® Xn)) ~ O.

(c) [x,y] =:$- L::=l[x,y](k) ~ o. (d) [X,jN,n(a)] = 1;; L:~[x(k), a ® IN-n] ~ O .•

2.3) Corollary: The closure N of Q(P(A(o))) in AN is a unital commutative C*algebra which commutes with all quasilocal observables a E Ax>' 2.4) Definition: N is called the algebra of all intensive observables.

We can view X to be the strong limit of N identical, mutually independent observables. By the symmetrization procedure all local effects disappear. Obviously N is generated by {Qx : X E A (standard) }.

Let X = S(A) be the state space of A, X = {p E A' : p(l) = Ilpll = I}. It is a w*-compact set. For p E X let p' = p®N denote the N-fold tensor product of p. Then P'p' = p'.

By p(a) = O(p'(a») there is defined a state on AN. By 2.2 (b) we obtain p'(xy) ~ p'(P(jN,2(X ® y) = p(x)p(y).

This implies that piN is multiplicative which proves the first part of the following theorem, due to Stl1lrmer [ST].

2.5) Theorem: N is C*-isomorphic to the C*-algebra C(S(A» of all continuous functions on the state space of A.

232 M. Wolff

Proof: Denote by fx the continuous affine function p -t p(x) on SeA). For rEIN (standard) and x!, ... Xr E A set SOQ(P(XI ® .. ®Xr ® IN-r)) = fXI . ·fxr·

So is well-defined and uniquely linearly extendable to a map S : Q(P(A(o))) -t C(S(A)) which is obviously multiplicative and contractive, satisfying S(R*) = S(R)* for all polynomials R. Hence S prossesses a unique extension to an equally denoted C*-homomorphism of N into C(S(A)). But {fx : X E A.a} separates points on SeA)). By Stone - Weierstrass's theorem S is onto.

Finally x -t x(l) -t x = P(x(l)) -t Q(x) is norm-continuous from A into N. Moreover it is linear. So every point ~ in the Gelfand space K, say, of N defines apE SeA) by p(x) = Q(x)(~). This shows that p -t p is surjective hence S is injective .•

2.6) From now on we identify Q(x) and fx for all (standard) x E A. We want to show that the C* -algebra generated by Aoo and N in AN is isomorphic to Aoo ® N.

To this end we denote by PK(K E *IN) the conditional expectation corresponding to the action of the permutation group S(K) on AK •

2.6.1) Lemma 1: Let x E * A fin be arbitrary, and choose n E IN standard. Then PN(jN,I(X))::::! In ® PN-n(jN-n,I(X)).

Proof: The lefthand side u is equal to j, L:~=l x(k), the righthand side v equals

N ~ n L:~=n+1 x(k). Hence

Inn N

U - V = N t;x(k) + N(N _ n) k~l x(k).

Since n is standard u - v ::::! O .•

2.6.2) Corollary: Let Xl, . .. , Xr E * Afin be arbitrary, where rEIN is standard. Moreover let n be a standard positive integer. For an arbitrary x in A set

I N

X = N Lx(k), k=l

N-n X = N ~ n L x(k).

k=l

Then Xl ... xr ::::! In ® Xl ... xr

Proof: Since l' and n are standard Xl' 'Xr ::::! II/=I(ln ® XI) by the lemma .•

In the following let n E N be standard. For y = Xl . . xr in An we set y = Xl . ,xr in

AN-n'

2.6.3) Lemma 2: Let a!, ... ap E * An,fin; let moreover YI E AN be of the form YI = II]!: 1 X j,I, where x j,1 E * A fin· Then L:i a,y, ::::! L:i a, ® YI

Proof: Since p is standard the assertion follows directly from 2.6.2 .•

2.7) Now we are able to prove the following theorem. Its importance lies in the fact that we obtain a structure which is independent of our special enlargement and our


particular N, but which is nevertheless something gained by a "generalized" limit process.

2.7.1) Theorem: The CO-algebra generated by AcX) and /If in.AN is CO-isomorphic to Aoo 0/1f = C(S(A), Aoo).

Proof: The last equality is well-known. In order to prove the theorem let us consider the set E of elements of the form L:i alhl, where all al are in A(oj, hence in some An for an appropriate n, and where hi = IIJ=dxj,! for some Xj,l E A.

E is a norm-dense linear subspace of the CO-algebra generated by Aoo and /If in .AN. Likewise the set F of elements of the form L:i al 0 hi is a dense linear subspace of Aoo 0/1f.

Choose such an element w = L:i al 0 hi, where ai, n, and hi are as above. Theorem 2.5 holds for all N as long as they are infinitely large. Thus we may assume that fXj,! = Q(Xj,I), where Xj,1 is as in 2.6.2 and Q : AN-n,Jin -+ .AN- n is the corresponding quotient mapping. Then w = Q(L:i al 0 tit) where ih = II~!xj,l' But by 2.6.3 w = Q(L:i aIYI), where YI = II~!xj,l' Since Q is a homomorphism w = L: alhl. It follows that the map U : L: al 0 hi -+ L: alhl is an isomorphic homomorphism of involutive algebras. So it may be extended to a CO-isomorphism from A 0 /If onto the C" -algebra generated by Aoo and /If in .AN .•

2.7.2) Corollary: There exists a conditional expectation E from A0/1f onto 10/1f satisfying E(X1 0 ... 0 x r) = 10 fx, . -Ixr (where Xl 0··· 0 Xr is identified with its canonical image in Aoo).

Proof: Let B denote the external involutive algebra generated by A(o) and P(A(o) in AN and set Eo = QPIB. Then the continuous extension of this map to all of A 0 /If is the desired E.

3 Dynamics in meanfield systems

Since the fundamental paper of K. Hepp and E.H. Lieb ([HL]), research on meanfield dynamics has made substantial progress; see in particular by [B01], [RI], [UN], [DW1], and [DW2].

In the following we want to give an easier proof of one of the main results of [DW2] (see theorem 4.6.). To this end we give first of all the relevant definitions and auxiliary results.

3.1) Definition: B E C( AN) is called a meanfield operator if B has the following properties:

(1) P(B):= frr L:"ES(N) U;l BU" ~ B (i.e·IIB - P(B)II ~ 0) (2) IIBIlIN is limited but ¢ O. (3) For every t ~ 0 exp(tB) = Tt is a completely positive unital operator.

Now denote by X. Y the (external) involutive subalgebra generated by X and Y.

234 M. Wolff

3.2) Definition: A meanfield operator B is called a meanfield generator if

(1) Tt(A(o)eP(A(o»)) C A(o) e P(A(o/ (the S-closure with respect to the internal norm on AN). (2) (Tt ) is strongly S-continuous on (A(o) e P(A(o»)).

3.3) Remarks: 1) PB ~ B implies PB ~ BP and a fortiori PTt ~ TtP.

2) Let B be a meanfield generator. Then, by setting Ttu = (~) for all u in the algebraic tensor product A(o) 0 N, there is uniquely defined a strongly continuous semigroup of completely positive unital operators Tt (t ~ 0) on Aoo @ N. The generator of this semigroup is closely connected with B, as we shall see later on. This justifies our definition of a meanfield generator.

3.4) Examples: 1) Let Q(6, . .. , er) = er1 ••• e~r be a monomial in r variables and let Xb . .. , Xr E Asa (standard) be arbitrary. Then H = N· p(x~al . @x~Ctr @IN-!Ct!) is selfadjoint (Ial = Laj), and is denoted by HQ(Xb ... ,Xr). It is easy to check that by the commutator bracket BQu = i[H, u] there is defined a meanfield operator BQ. Now it is clear how to define BQ for an arbitrary (standard) polynomial Q in r variables with real coefficients and for each standard r-tuple (Xl, ... , Xr) E A~a. These operators were the starting point of the theory of meanfield systems. They are in fact generators in our sense as we will see later on. 2) Let G E .c(Ag) (g and G standard) be such that exp(tG) is unital and completely positive for t ~ o. Then B = NP(G @ IN-g) is a meanfield operator. This kind of operator corresponds to the polynomially bounded meanfield operators of [DW1] , [DW2], so we shall use the same expression for our ones. The example (1) fits into this framework. Indeed, set h = X?Ctl •.. @ x~Clr and G(u) = i[h, u] on Ag (g = lal). Then BQ = NP(G @ IN-g).

In the next section we shall show that a meanfield operator as in example (2) is always a meanfield generator. The corresponding meanfield dynamics (1't ) (see 3.3) leaves 1 @C(S(A)) invariant and there it is induced by a classical flow, which is best understood in the case where A is finite dimensional. The rest of this section contains the connection between internal one-parameter semigroups and their nonstandard hull which we need in the rest of the paper.

3.5) Proposition: Let B be an internal, internally bounded linear operator on the internal Banach space E, and assume that there exists a standard constant M, such that IletBl1 ::; M for all t ~ O. Let F C EJin be the subspace of all x such that o ::; t -+ etB x is S -continuous at o. Then F is invariant under Tt : = etB (t ~ 0), and by Ttx = (~) there is defined a strongly continuous one-parameter semigroup (1't)t>o on F = F/(Eo n F), which is norm bounded by M.

Proof: Straigtforward (cf. also [W], sect. 1 and prop. 4.1) .•

We describe the generator of (Tt):

3.6) Corollary: The generator A of (Tt) is given by 1>(A) = [(AI - Btl]"(F) for

some,x > 0 (standard), and Ax = (i3;) for all x E 1>(A).

Proof: (I) Let ,x > 0 be standard. Then (AI - Btl =: R(,x) = 1000 e->.tTtdt


(by transfer) hence IJR(A)IJ $ If, that is R(>\)E C E. Moreover for y E F and x = R(A)y we have

t •• • Ttx = e>.t(x - fo T.yds) for all t, which shows that x E F, hence R>.(F) C F.

(II) Let y be in F. Then T.y RJ Y implies t f~ T.yds RJ y for all t RJ 0, hence

Ttxt- x RJ AX - y for all these t. Since our model is saturated, we conclude that to every c > 0 standard there exists J > 0 standard satisfying II(Ttx-x )/t- (Ax-y)11 < c for all 0 < t < J. This implies x E V(A) and Ax = AX - y = (13;). (III) By II (A - A)R(A) = -I, and the corollary follows .•

3.7) Corollary 2: F = {x E EJin : Bx E EJin}

Proof: By the transfer principle we have Ttx - x = f~ T.Bxds, hence for all t RJ 0 IITtx - xii ($ t· M ·IJBxll) RJ 0 holds iff IIBxl1 is limited .•

4 Polynomially bounded meanfield operators

In the following let 9 E IN be a fixed standard integer and let G E C( Ag) be a standard, bounded linear operator on Ag such that for every t ;::: 0 exp(tG) is unital and completely positive. This implies in particular Gl = O.

First of all we want to establish another formula for B = NP(G@IN_g). To this end let L C {1, ... ,N} be a fixed set of size g, and denote by <PL the unique permutation mapping L onto {1, ... g} =: K and N\L onto N\K in such a manner that it is order preserving on L and order preserving on N\L. Let 71' be another permutation mapping L onto K. Then there exists a unique permutation p on K and a unique permutation a on N\K such that 71' = (p EEl a) 0 <PL where p EEl a is the permutation acting as p on K and as a on N\K. Let i denote the identity (on the set under consideration). Then p EEl a = (p EEl i)(i EEl a) = (i EEl a)(p EEl i). so for Q = G @ IN-g we have

Setting - 1 '" -1 G=,. ~ Vp GVp

g. pES(g)

where Vp is the automorphism on Ag induced by the permutation p we obtain

B = g!(N - g)! '" U-10u. (N - i)! ~ ~L- ~L'

ILI=g

Denoting Ui{t2V"'L by GL we can summarize our considerations as follows:

236 M. Wolff

4.1) Lemma: B:= NP(G i8l IN-g) = -rN) 2:: GL. 9 ILI=g

It follows in particular that we can assume G = G without loss of generality whenever it is convenient. We put together some easy consequences of this lemma:

4.2) Lemma: Let K = {1, ... , r} for some standard r ;::: g} and set for j = 0,1

Bi = -rg

) 2:: G L. Then the following assertions are true: ILI=g

ILnKI=i

(1) liB - (Bo + Bdll ~ 0

(2) IIBIII ~ r· 9 ·IIGII (3) Bo = 0 on * Ar (identified with jN,r( * A)) (4) IIB1PI*Ar ll ~ 0

Proof: From 4.1 we obtain B = -rN) t 2:: GL · So we first have to show 9 )=0 ILI=g

ILnKI=i

116 l12g GLII ~ 0 for j ;::: 2. Now for 0 :5 j :5 9 there are S(j,g,r) =

ILnKI=i

C) (; =;) subsets L ofsizeg satisfying ILnKI = j. Moreover IIGL(a)11 :5 IIGllllal1

independently of L as follows easily from the definition of GL. Finally

S(j,g,r)' N

(~)

(j)(N - r)!g!(N - g)!

(g - j)!(N - r - 9 + j)!(N - 1)!

(j)g! (g - j)!

-. t(j,g,r).

(N - r) . ·(N - r - 9 + j + 1) (N-1)··(N-g+1)

, , If j;::: 2 then t(j,g,r):5 (N ~';'+ 1) ~ 0, which proves (1).

If j = 1 then t(j, g, r) ~ rg which implies 2. If a E • Ar and a = a i8l1N-r then for L n K = 0 U¢L (a) = l r@b for some bE A N- r. But Gl = 0, hence GL = 0, so (3) follows. It remains to prove (4). Claim: If b = P(ci8llN-r) is limited then

b ~ -r1) 2:: U4>( ci8llN -r) =: b. For considerations similar to those of lemma r 4>,4>(K)nK=0

4.1 show


2: U4>L(C@ IN-r) 1 r

= -2: e) k=1 ILI=r ILnKI=k

Now S((6,r) R:;j 0 for k ;::: 1, so the claim is proved. But since Bl is limited we

have B1 P(c@ IN-r) R:;j Bl(b) = 0 since GI = O .•

So far we have all the ingredients needed to prove that B behaves almost as a derivation - at least for a restricted set of products:

4.3) Proposition: (a) Let al, ... ar E * AJin be arbitrary (r: standard). Then r

B(IIj=laj(j)) = 2:IIk#jak(k). B(aj(j)) j=1

(b) Let r ;::: g be standard and let a, b E oAr standard. Set a = a @ IN-r, C = P(b@IN-r). Then

(1) Bo(ac) = aBo(c) (2) Bl(ac) = Bl(a)c

(c) Let r,s be standard, and let a E An b E A. be standard and c = P(b@ IN-.). Then B(ac) R:;j B(a)c + aB(c). (d) Let a,b as in (c) and set d = P(a@IN-r), c = P(b@ IN-.). Then B(cd) R:;j B(c)d + cB(d).

Proof: (a) Using 4.2 we obtain for a = II~aj(j) N r

B(a) R:;j Bl(a) = Fg) ~ 2: U~L1GU~L(a). LnK={k}

ILI=g N ow if I L I = g and L n K = {k} (1 S; k S; r) then Ui:GU~L(a) = II/#a/(I)Ui:(G(ak @ Ig-d)· Inserting this into the formula above we obtain the result.

(b) (1): Since U~L(C) = c we obtain Bo(ac) = (hg) 2: Ui:(G((U~L(a))c)). ILI=9

ILnKI=0 By the formula for P in the proof of 4.2 (4) we get

c = P(b@ IN-r) = fry 2: U~L,(b@ IN-r)' r IL'I=r

If LnK = 0 = L'nK then U~L(a)U~L,(b@IN-r) = Ir@dforsomed E AN- r hence G vanishes on such an element. Moreover Ui:(a) = Ir @ dL for some dL E AN- r whenever L n K = 0. This alltogether gives

Bo(ac) = a(~) 2: Ui:(G((~) 2: U~L,(b@IN-r))) 9 LnK=0 r IL'I=r

= aBo(c).

238 M. Wolff

(2): Since B1 is norm-limited we can use the approximation of p(b) in the proof of

4.2 obtaining B1(iic) ~ (N)N(N) 2: 2: Ui;(G((U~L(ii))U~L,(b))). 9 r ILnKI=l L'nK=0

A careful but straightforward calculation similar to that one for Bo above will give the formula. (c) Let t = max(r,s,g). Then one may view a,b as elements of At and G as an operator on At (replacing G by G ® It- g if necessary). But then 4.2 and (b) will give

B(iic) ~ Bo(iic) + B1(iic) ~ iiBo(c) + B1(ii)c ~ iiB(c) + B(ii)c,

where Bo, B1 are built with respect to K = {I, ... t}. (d) Apply the conditional expectation P to (c) and notice PB = BP .•

We need one further lemma before we can prove our first main theorem.

4.4) Lemma: For r, s E IN standard denote the S -closure of the external Q;-linear hull of {a ® IN_rP(C® IN-.): a E A, c E A., standard r} by £r, •. Then B maps

A (identified with jN,r(A)) into £r,g-l and IIBIAril ~ IIGIIg· r.

Proof: We build Bo, B1 as in 4.2 and obtain BIAr = BdAr by 4.2 (3), hence the norm estimate follows from 4.2 (2). Since B1 is norm-limited we need only to prove B1(IIj=laj(j)) E £r,g-l for all at, ... ar E A standard. By 4.3 (1) we can reduce this further to B1 (a(j)) E £r,g-l for all 1 ~ 9 ~ r and a E A. But

N r

B1(a(j)) = (~) ~ L Ui;QU~L(a(j)), LnK={k}

ILI=g

where G = *G ® IN-g' Since cf>r;1(l) ~ r + 1 for 1 < I ~ rand cf>r;1(1) = k we obtain

L Ui;Q(a(l)).

ILI=g ILnKI={j}

Now define for 1 ~ k ~ N \rk: * A -t AN by \rk(a) = a(k)(= 1k-1 ® a ® IN-k) and for L' = Lj\j Vu = ®kEL'Vk: *Ag-1 -t AN. Then Ui;(Q(a(1))) = (Vj ® Vu )(G(a(l)). Using 2.6.1 we obtain

B(a(j)) ~ (Vj ® PN-1)Q(a(1)) E Ag• P(A(9-1/ (S-closure) .•

4.5) Corollary: Let £r,oo be the S -closure ot the external Q;-linear hull of {a ® In-r . P(A(o») : a E Ar standard} or in other words: £r,oo = Q-1(Ar ® N). Then B1 maps £r,oo into itself.


Proof: Bl is limited, hence S-continuous, and the assertion follows from 4.4 .•

Now we are able to prove the following final theorem on polynomially bounded meanfield operators which is essentially due to N. Duffield and R. Werner [DW2]:

4.6) Theorem: Let G E £(Ag) be a continuous linear (standard) operator on Ag such that for all t ~ 0 exp( tG) is a unital completely positive operator on A g. Then B = NP( G) is a meanfield generator. More precisely set Tt = exp(tB) for t ~ O. Then for every standard t, Tt leaves invariant V := Ax, IZl N and the restriction of the semigroup (Tt) to V is strongly continuous. Moreover the following assertions are true: (1) ETt = TtE for all t ~ 0 and the restriction of (Tt) to 1 IZl N = E(V) is a strongly continuous semigroup of C·-homomorphisms, thus it is induced by a flow <Il t on S(A) (notice N = C(S(A))). (2) For every r ~ 9 (Tt) leaves invariant Ar IZl N. 4.7) Corollary: Let A denote the generator of (Tt)lv, Then the following assertions hold: (1) The linear hull Vo of the set A(o) . E(A(o» = {af : a E A(o), f E E(Ao)} ~s contained in the domain V(A) of A. (2) For all aI, ..• ar E A

r

A( al IZl .. lZlar) = L IZlj#aj . A( ak IZll) k=1

(modulo the obviously necessary identifications). (3) For all rEIN, a E Ar and f = E(b) E E(A(o» A(af) = A(a)f + aA(f).

Proof of 4.7: This follows from 4.3, 4.6 and 3.6 .•

Proof of 4.6: Let Fe AN,fin be the space on which (Tt ) is strongly S-continuous (cf. 3.5). Then by 3.7 and 4.2, 4.3, 4.4 the external <V-linear hull of {a IZlIN-r' b: r standard, a E Ar standard, b E P(Ao)} lies in the domain of the generator A of (Tt) on F(= Q(F». Moreover 4.4 and the equation PB = BP implies that IIBIP(Ar)11 :::; rgllGII· (I) Claim: N c F and Tt(N) c N Proof. By the last inequality and 3.6 we obtain Q(P(Ar» := Vr C V(A) and IIAIVrl1 :::; rgllGII· Since A is closed Vr C V(A) and IIAlvJ :::; rgllGIl. By induction on n we obtain for a E Vr II Ana II :::; gnIlGllnII~:~(r + kg)lIali. Let r' = min{k ~ rig: k E IN} Then

II~:~(r + kg) < gnII~:~(r' + k) = gn . n! (r' ~ n)

< gn. n!2rl .2n.

00 kAk This shows that for 0 :::; t < (IIGII . g2 . 2 . 2r >-1 the series {; y. converges, i.e.

a is an analytic element in V(A), and moreover the series is equal to Tt(a), and lies in N. Since U::l Vr =: Voo is dense in N, the assertion follows.

240 M. Wolff

(II) Fix an rEIN, and consider Bo and Bl as in 4.2. Bl is limited. Set V; =

etB1 • By 3.4 and 3.5 Vt is uniformly continuous on AN. Moreover 4.5 tells us that Vt(Ar @/If) cAr @/If. Now consider also St = exp(tBo) Bo(Ar) = 0 by 4.2 (3) and Bolp(Aol ~ Blp(Aol by 4.2 (4).

This gives StlArl8iN = IAr @ TtIN, possessing the generator Ao = IAr @ AINnV(Al. But then A2 = Ao + Ai is a generator, where Ai = 81 is bounded. The semigroup generated by A2 is given by Wt = liII1n-too(St/n· Vt/n)n ([BR1], 3.2.31).

(III) Claim: A2 = AID(A) n Ar @/If.

Let a E Ar, bE P(A.) be arbitrary. Then (ab) = a@b E D(A) since B(a@ IN-r· b) is finite. Moreover 4.3 (b) and 4.2 (1) imply l}(a@ 1N-r· b) ~ B1(a@ 1N-rb) + a@ IN_rBo(b). Thus A(a@b) = A 1(a@b)+Ao(a@b). But B1(a@IN_r)b ~ B1(a@IN_rb), by 4.3 b (2). So the assertion follows easily.

(IV) Finally we have to prove that TtlN consists of *-homorphisms. Since /If is commutative, and Tt maps the selfadjoint part onto itself it is enough to show that Ao is a derivation. But this follows from 4.3 c .•

4.8) Remarks: 1) As was pointed out already, theorem 4.6 and its corollary are essentially due to N. Duffield and R. Werner [DW2]. We have presented them here since our proof seems to be much easier. Moreover our approach reflects the physical intuition: Without the use of the equivalence relation " ~ ", i.e. without the use of the quotient AN, we remain totally within the microscopic picture. Only the use of that equivalence relation yields the result. But this relation means "indiscernable by experiments of a macroscopic type". Loosely spoken: classical effects result from quantum mechanical microscopic systems by observation methods suppressing the microscopic details or making them indiscernable. 2) One of the concrete results coming out of 4.6 is the following observation due to Unnerstall [UN] : even if the given local action G is dissipative the induced dynamics on the "classical" part /If is "deterministic", i.e. it comes from a flow, a result which supports again the macroscopic nature of the classical dynamical part.

5 A pproximation of meanfield generators

Let £ denote Q-l(Aoo @/If), and let B be a meanfield operator. An S-dense subspace Me £ is called a core of B, if {(x, Bx) : x E M} is S-dense in {(x, Bx) : x E D(B)} where D(B) = {x E £ : Bx is limited}.

First of all we formulate the following "uniqueness theorem ":

5.1) Proposition: Let B E C(AN) be meanfield generator, and let D E C(AN) be arbitrary such that for A > 1 R(>., D) exists and is limited and moreover such that Du ~ Bu for all u in some fixed core M of B. Then D is also a meanfield generator and both induce the same meanfield dynamics on Aoo @ /If.


Proof: Since M is a core and 1 is in the resolvent set of B (A - B)(M) is S-dense in E for all standard A 2::: 1. Now R(A, D) - R(A, B) = R(A, D)(B - D)R(A, B). Hence R(A, D)x ~ R(A, B)x for all x E (A - B)M. Since both operators are limited this approximate identity holds on E, hence R(;::D) = R(;::B) on AJO 0.Af and the assertion follows .•

Now we want to calculate explicitly the action of a polynomially bounded operator B, induced by a hamiltonian. Let R = R(6, . .. , ~.) be a polynomial in s variables.

Then the partial derivatives DjR = ~ are again polynomials. Now if Xl, . .. , x.

are elements of Asa then we obtain R(Xl"'" x.) by replacing ~j by Xj' We denote this polynomial by R(i). If R(i) = xrl . ·x~' then we define R(i) by x?al 0· . x~a, as in section 2. For a general polynomial R(i) we define R(i) by linear extension. The element P(R(i)) will be denoted by hR. Clearly Q(hR) = R(fxJ)' .. , Ix,) (see section 2). If R is a monomial, R(i) = xrl···x~', then we define GR on Ag(g = 2:0j) by G R (a) = i [R( i), a], and similarly for general polynomials. Then NP(GR) = B is given by B(u) = Ni[hR,uJ (see 3.3) thus B is an internal inner derivation.

The next two formulas are easily proved by showing them at first for monomials:

5.2) Lemma: •

(a) For a E A B(a(l)) ~ i 2:[xj,a](l)· h~ j=1 J . --

(b) For a = P(a(l)) B(a) ~ i 2: [xj,aJ· h~ j=1 J

Proof: (a) Let R(6, ... ,~s) = 6···~. and choose XI, ... Xg E A.a not necessarily distinct.

Then hi? = P(xl(l)··· xg(g)) = (N N/)! ~(O)xl(kl) . ,xg(kg) (see sect. 2). This k

gives

B(a(l))

Here the roof denotes as usual that this factor does not occur and k' denotes all those indices where kj does not occur. But this gives (cf. sect. 2)

9

B(a(l)) = i ~)xj, aJ(l) 0 PN - 1(Xl(1) . . x7i) . ·xg(g)) j=1

9

~ i ~)Xj, aJ(l)· hDjR j=1

242 M. Wolff

and (a) follows easily. Now we apply P to (a) and obtain (b) .•

Because B is an internal inner derivation, it is completely determined by formulas (a) and (b) (cf. also 4.7). In particular we obtain the following corollary (see e.g. [BOI], [DWI]).

5.3) Corollary 1: The dynamics induced by R(Xb"" Xg) on./lf = C(S(A) is completely described by the following Poisson bracket for polynomials S(fYI' ... ,fy.) :

{R(fxJ'" .. fXg), S(fy!>" .. , fy.n = L DkR(fx!>" .. , fxg)D1S(fyJ'"" fy.)f[xk, yd k=1..g 1=1. .•

By 4.6 and 4.7 the global dynamics on Aoo {l)./If leaves invariant all Ar {l)./If whenever r ~ g. But since in the case under consideration B is a derivation this holds also for all r. Since we obtain B on Ar e P(A(o)) as a sum of operators acting on A{I}eP(A(o)) we need only to consider this case. Lemma 5.2 yields that BIAI.P(A(o))'

• can be decomposed into a bounded operator BI : a(l)· u -+ i L[aj,a](l)· hD -R • U

j=I J

and an unlimited one. Going into the quotient we obtain the following corollary (A{k} = {lk-I {l) x : x E A} C Ale = A®k) :

5.4 Corollary 2: (Tt) leaves invariant all sub-algebras A{k} {l)./If. It is uniquely determined by the action of its generator on Al {l)./If, which is given by AI1J(A)nAI®N =: Al = A2 + Ao where Ao(a(l) {l) f) = L[Xj, aj {l) DjR· j, i.e. Ao is bounded, and A2 = I {l) A3, where A3 is uniqueley determined by the Poisson brackets for polynomials of coroll. 1.

We come now to the final result of this section. For the sake of convenience we assume that A is finite-dimensional. Then we may select a base xl, ... X. of A.a SlRi over lR which is orthonormal with respect to the trace (we consider A to be a C*-subalgebra of some full matrix algebra). Then since A.a is a Lie-algebra with respect to [.,.J, SeA) becomes a Poisson manifold in a natural manner (see [ARJ, p. 457. All these cases of a Poisson structure occur in [BOlj and are already mentioned in [HLj.). Moreover all polynomials R(YI,' .. yg) are expressable as polynomials in the usual sense, refering only to {Xl, ... ,x.} (see also [UN]). The Poisson bracket is given for CI-functions F, G by

(see [ARJ, p. 457). So every C2-function F defines a flow <JlF on V'.

In the following we suppose that F is a C2-function such that SeA) is positively invariant set of <JlF . We denote by (St) the semigroup of*-automorphisms on C(S(A)) defined by (Sd) (p) = f ( <Jl ( t, p) ). Since F is of class C2, <Jl F is of class C 1 with respect to p E SeA), hence CI(S(A)) is a core of the generator of (St).

5.5) Theorem: Let (Rn) be a sequence of real polynomials which converges to F uniformly with all its derivatives of order one. Let (Tn,t) be the meanfield dynamics


on A", 129 N corresponding to !In (see 5.3). Then there exists a strongly continuous semigroup crt) of *-homomorphisms on Aoo 129 N such that Tn,tu -+ Ttu uniformly on every bounded interval. The restriction TtlN is given by the flow induced by F on SeA). Roughly spoken: every C 2-function F which induces a Cl-semiflow on SeA) via the Poisson bracket above determines a classical dynamics which is the restriction of a meanfield hamiltonian dynamics on Aoo 129 N.

Proof: By 5.3, 5.4, 4.6 and 4.7 we have only to prove the convergence on Al 129 N. We decompose the generators AnIV(An) n Al 129 N into A2,n + Ao,n according to

5.4. Then (Ao,n) converges strongly to a bounded operator Ao, say, on A I 129N since DjRn -+ DjF uniformly by hypothesis. A2,n = I 129 A3,n, where A3 ,n is given by the Poisson bracket {!In, .}. This bracket converges strongly to {F,.} on C 1(S(A)) by hypothesis. The latter is a derivation G -+ o( G) = {F, G} the closure of which is the generator of (St) (see above and note that C1 is a core of 0). But then (Sn,t) -+ (St) strongly (uniformly on compact intervals), (where (Sn,d corresponds to {Rn ,'} = On = A3,n) by 5.7.11 in [FAT]. So A2,n + Ao,n -+ I 129 0 + Ao on Al 8 C1(S(A)) (where 8 denotes the algebraic tensor product) which is a core of I 129 0 + Ao since Ao is bounded by the uniform boundednes theorem. 5.7.11 in [FAT] yields our theorem .•

The proof has shown how the generator of the limit looks like:

5.6) Corollary: The generator A of the dynamics of 5.5 is given on AI8C1(S(A)) •

by A(a(l) 129 G) = j~ [xiJ a](l) 129 (DjF) . G + a(l) 129 f.t DjF DkGf[xj, xkj'

5.7) Final remarks: 1) Obviously A(o) 8 C1(S(A)) is a core of A. Since A is a derivation it is completely determined by its values on Al 8 Cl(S(A)). 2) Let R be a polynomial such that SeA) is positively <1>R - invariant. Then the classical dynamics on C(S(A)) given by the Poisson bracket is nothing else than the "statistical limit" of the quantum dynamics given by the corresponding commutator

bracket [NhR,'J on P(A(o)( This establishes an idea of P. B6na4

The more common idea that the transition from quantum mechanics to classical mechanics is given by letting Ii be standard (:rf 0) or infinitesimal (as is sketched in [00]) seems to be more restrictive. 3) Within our framework it is possible to extend the correspondence of {R,·} and [NhR,'j from polynomials to arbitrary C2-functions: let F be approximated by the internal polynomial R, say, of degree K (K ~ 00), i. e. II * F - RII + L II • DjF - DjRII ~ O. Now choose N = K! (or N = KK). Then (cf. the paragraph preceeding 5.2) NP(GR) is given by Ni[hR,·J = B which gives B(a) ~ i L~ [xj, aJhiJ:R. So even in this case the correspondence between the two

J

types of brackets, i.e. between classical and quantum mechanics is established. We do not want to go into the details.

40ral discussion during my stay at Bratislava in January 1993, cf. also [B02]. I would like to thank the Komenius University at Bratislava for the kind hospitality.

244 M. Wolff

References

[AFHL 1 S. Albeverio, J. E. Fenstad, R. H¢egh-Krohn, T. Lindstr¢m: Nonstandard methods in stochastic analysis and mathematical physics, Academic Press, Orlando etc. 1986.

[AR 1 V.1. Arnold: Mathematical methods of classical mechanics, second ed. Springer Verlag Berlin etc. 1989.

[Boll P. Bona: The dynamics of a class of quantum meanfield theories, J. Math. Phys. 29 (10), 2223 - 2235, 1988.

[B02 1 P. Bona: Quantum mechanics with mean - field backgrounds, Comenius University, Fac. of Math. a. Physics, Physics Preprint No. PH10-91, Oct. 1991.

[BRl 1 o. Bratteli, D. W. Robinson: Operator algebras and quantum statistical mechanics I, Springer Verlag Berlin etc., 1979.

[DR 1 E. Duffner, A. Rieckers: On the global dynamics of multi - lattice systems with non-linear classical effects, Z. Naturforsch. 43a, 521 - 532, 1988.

[DWl 1 N. G. Duffield, R. F. Werner: Mean-field dynamical semigroups on C*algebras, Rev. Math. Phys. 4, 383 -424, 1992.

[DW2 1 N. G. Duffield, R. F. Werner: Local dynamics of mean-field quantum systems, Helv. Phys. Acta 65, 1016 - 1054, 1992.

[FAT 1 H. O. Fattorini: The Cauchy problem, Encyclop. of Mathematics and its Appl., Addison-Wesley Publ. C., London etc. 1983.

[GVV 1 D. Goderis, A. Verbeure, P. Vets: Non-commutative central limits, Probability Theory, ReI. Fields 82, 527 - 544, 1989.

[HL 1 K. Hepp, E. Lieb: Phase transitions in reservoir-driven open systems with applications to lasers and superconductors, Helv. Phys. Acta 46, 573 - 603, 1973.

[HM 1 c. W. Henson, L. C. Moore: Nonstandard analysis and the theory of Banach spaces, in: A. E. Hurd (ed.): Nonstandard analysis - recent developments, Lecture Notes in Mathematics Vol. 983, Springer Verlag Berlin etc. 1983, 27 -112.

[HULO 1 A. E. Hurd, P. A. Loeb: An introduction to nonstandard real analysis, Academic Press, Orlando etc. 1985.

[MS 1 G. Morchio, F. Strocchi: Mathematical structures for long-range dynamics and symmetry breaking, J. Math. Phys. 28,622-635 (1987).


[MSTV 1 J. Manuceau, M. Sirugue, D. Testard, A. Verbeure: The smallest C"algebra for canonical commutation relations, Commun. Math. Phys. 32, 231, 1973.

[N 1 E. Nelson: Internal set theory, Bull. Amer. Math. Soc. 83, 1165 - 1198, 1977.

[00 J I. Ojima, M. Ozawa: Unitary representations of the hyperfinite Heisenberg group and the logical extension methods in physics, Research Institute for Mathern. Sciences, Kyoto University Kyoto, Japan, RIMS-899, Sept. 1992.

[P J G. K. Pedersen: CO-algebras and their automorphism groups, Academic Press, London etc. 1979.

[RI J A. Rieckers: On the classical part of the meanfield dynamics for quantum lattice systems in grand canonical representations, J. Math. Phys. 25, 2593 -2601, 1984.

[RW 1 G. A. Raggio, R. F. Werner, Quantum statistical mechanics of general meanfield systems, Helv. Phys. Acta 62, 980 - 1003, 1989.

[SL J K. D. Stroyan, W. A. J. Luxemburg: Introduction to the theory of infinitesimals, Academic Press, New York etc., 1976.

[ST 1 E. St~rmer: Symmetric states of infinite tensor products of CO-algebras, J. Funct. Anal. 3-, 44 - 51, 1969.

[TA 1 M. Takesaki: Theory of operator algebras I, Springer Verlag New York etc. 1979

[UN J T. Unnerstall: The dynamics of infinite open quantum systems, Lett. Math. Phys. 20, 183 - 187, 1990.

[WH 1 M. P. H. Wolff, R. Honegger: On the algebra of fluctuation operators of a quantum meanfield system, Preprint Univ. Tiibingen 1993.

Index 2-dilation, 4

algebra C*-, 228 differential, 60, 112 of generalized functions, 59 of intensive observables, 231 of local observables, 230 quasilocal, 228

analytic functional, 108 angular cutoff, 223 approximate symmetric sequence, 229 approximation

hyperfinite, 37 polynomial, 103

Banach-Mazur limit, 4 bracket

commutator, 234,243 Poisson, 242, 243

Brownian bridge, 176, 180 nonstandard, 181

Brownian motion, 143, 144, 150, 151, 176, 186

Anderson's, 144, 180, 186 Levy's, 176

capacity, 48 Carleman system, 71 Cauchy problem, 69, 72, 108, 207

with singular data, 70 chaos decomposition, 154 cocycle, 201

compact, 202 generalized, 201 regular, 201 singular, 201

co cycle property, 200 generalized, 202

commutator bracket, 234, 243 conditional expectation, 87, 233 continuous, M-, 103 convergent,~-, 125

247

countable neighborhood property, 130 cutoff

angular, 223 cycle, hyperfinite, 203

decision problem, 213 delta wave, 70 derivative

directional, 159 regularized, 71

differential algebra, 60, 112 differential operator, 68

nonlinear, 68 diffusiol!, 143, 144

on manifolds, 143 Dirac measure, 57 distribution, 56, 107

generalized, 107, 113 generalized localizable, 108 localizable, 119 multiplication of, 56 Schwartz, 107

dynamical system, 200 singular stochastic, 200

dynamics, 229 in meanfield systems, 233 local, 229 meanfield, 229 meanfield hamiltonian, 243

equation Boltzmann, 221

without angular cutoff, 223 delay, 201 elliptic, 205 Foias-,34 Korteweg-deVries, 71 Lewy's, 108,117,118 Navier-Stokes,20 of hydromechanics, 24 of magneto-hydro-dynamics, 24 of thermohydraulics, 24 partial differential, 57, 65, 107

248

nonlinear, 57, 65 of Cauchy-Kowalewski type, 72

quasilinear hyperbolic, 71 Schrodinger, 71 semilinear wave, 69 stochastic differential, 144,147, 167,

200 infinite dimensional, 205 nonlinear, 205 partial, 201

stochastic Navier-Stokes, 21, 27 equivalence relation, external, 122 ergodic

interval, 15 map, 201 state, 8 transformation, 171

exactness of solutions, 69 experiment, 212, 213

statistical, 212

fluctuation, 228, 229 Fourier series, 37 Fourier transform, 39

generalized eccentric (operator), 6 generalized functions, 56

algebras of, 59 as initial data, 69 nonlinear theory of, 56, 59

generator, 92, 167 meanfield, 234 of a Co-semigroup, 92 of the Ornstein-Uhlenbeck semigroup,

167 geometry, non commutative, 4 group

hyperfinite, 37 LCA, 37 locally compact abelian, 37

growth bound, 92

Haar measure, 39 Hadamard number, 98 hamiltonian, 241 harmonic analysis, 37

heat kernel, 143, 145 nonstandard, 143

Heaviside function, 58 hyperfunctions, 108

integrable SLP-, 134, 135 Bochner, 76, 81, 133

extended, 135, 136 Dunford, 82 Pettis, 76 S-Dunford, 83 S-Pettis,83 uniformly Pettis, 84

integral Bochner, 28, 76, 81, 136 indefinite Bochner, 84 indefinite Pettis, 84 Ito, 27, 150, 152, 161 Loeb-Osswald, 136 multiple Wiener, 151

representation of, 152 Pettis, 76, 83 Skorohod, 161 stochastic, 27, 150, 157

integral sequence, 5 integration

by parts, 163, 168 Ito, 151

intensive observable, 231 invariance principle, 176 Ito integral, 27 Ito integral, 150, 152, 161 Ito integration, 151

lifting, 75 S2-integrable, 39 p" 132, 135 SL2 , 151, 152 SLP,132 weak, 80

lifting theorems, 82 limit vector space, 126 local convergence, 102

set of, 102 localization, 61, 62

Index

Index

Loeb measure, 75, 132, 171, 178, 187, 203

on 'JR, 222 Loeb space, 75, 76, 132, 143, 150, 176,

206,214 filtered, 29

Lyapunov exponent, 201

Malliavin calculus, 149, 165, 195 Malliavin covariance, 159 manifold, 144

Riemannian, 144, 145 Maxwellian function, 223 meanfield generator, 234 meanfield system, 228 measurable, Pettis, 80 measure, Haar, 39 microcontinuity, 96

complex, 96 microcontinuous, 96

complex, 96 real, 103

monad, 47, 51, 124 71"-, 124 discrete, 124 filter, 124 of a topology, 124

multiplicative ergodic theorem, 200,202

Navier-Stokes equations, 20 nonstandard functional analysis, 23 nonstandard hull, 77

of a one-parameter semigroup, 234 weak, 77, 78

observable, 228 fluctuation, 228 intensive, 228, 231 local, 229

operator collision, 221 compact, 3 completely positive, 234 derivation, 149, 155 differential, 68

nonlinear, 68

discrete Malliavin, 198 divergence, 161, 168 eccentric, 6 generalized eccentric, 6

249

generalized Ornstein-Uhlenbeck, 150 gradient, 149, 155 hyperfinite Ornstein-Uhlenbeck, 196 Laplacian, 144 Lewy's, 110 linear partial, 107

elliptic, 109 regular, 109

local, 157 Malliavin, 149, 165 meanfield, 233 number, 166 polynomially bounded, 234, 241 pseudodifferential, 71 regular linear partial, 107 Skorohod integral, 149 trace class, 5

Ornstein-Uhlenbeck process, 167 infinite dimensional, 186, 187, 195,

198 Osedelec's subspace, 202

partial differential equation, 57, 65, 107 nonlinear, 57, 65

Poisson bracket, 242, 243 Poisson manifold, 242 polydisc, 98

closed, 98 polynomial

Hermite, 154, 167 nonstandard, 96, 98

quantum field theory, 4, 72

random walk, 143, 186 Anderson's, 143, 176, 208 hyperfinite, 176

regularity elliptic, 66 theory of, 66

Robinson-Principle, 124, 128

S-continuity, 96

250

S-continuous, 96 semigroup, 91, 92

Co-,91

set

nonstandard hull of, 234 of operators, 92 one-parameter, 91, 234 Ornstein-Uhlenbeck,167 strongly continuous, 92, 234, 243

irreducible, 49 saturated, 49 separating, 75 sober, 49 super-sober, 49

shadow space, 123 shift, 203

Kamae's universal, 203 shock wave, 67 solution

analytic functional, 108 classical, 58, 70 exactness of, 69 fundamental, 118, 119 generalized, 58, 70

unique, 71 uniquness of, 69

global, 72, 108 of Lewy's equation, 117 of PDE, 107 shock wave, 67 stability of, 69 statistical, 34 weak, 24, 25, 72

space df,131 gDF,131 locally compact, 46 Loeb, 75, 76, 132, 143, 150, 176,

206,214 of experiments, 212 shadow, 123 sip-, 177 sober, 49 state, 228, 231 super-sober, 49

spectral bound, 92 spectral mapping theorem, 92 spectrum, 94

approximate point, 94 Lyapunov, 201, 204

stability of solutions, 69 state, 7, 228

~-invariant, 10 2-dilation invariant, 7 Dixmier-type, 14 ergodic, 8 extremal, 8 generic, 7

Index

permutation invariant, 228 stochastic N avier-Stokes equations, 21,

27 stochastic process, 75

empirical, 180 generalized, 72 normalized sum, 179 Ornstein-Uhlenbeck, 167, 186 Pettis integrable, 88 Pettis integral, 88 Pettis measurable, 78

support of a generalized distribution, 114

theorem Gearhart's, 92 Keisler's Fubini, 86 of vanKampen, 40

topology Fell's, 46, 47 Lawson's, 47, 48 myope, 46, 48 Vervaats sup vague, 48

trace, 3,4 non-normal, 3 singular, 3-5, 8

transfer process, 70 transformation

ergodic, 171, 172 measure-preserving, 172

transition kernel, 213 generalized, 213

Index

truncation, 101 hyperfinite, 101

wave delta-, 70 shock, 67

weak solution, 25, 28, 72 weight, 4

normal,5 Wiener chaos, 152

nth, 152 Wiener measure, 149,151,177,196 Wiener process, 27

IH-valued, 27 Wiener-Ito chaos, 150, 151 Wiener-Ito chaos decomposition, 152

251

Advances in Analysis, Probability and Mathematical Physics: Contributions of Nonstandard Analysis

Documents

Transcript of Advances in Analysis, Probability and Mathematical Physics: Contributions of Nonstandard Analysis