Stochastic Analysis and Random Maps in Hilbert Space [Reprint 2018 ed.] 9783110618143, 9789067641630

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Stochastic Analysis and Random Maps in Hilbert Space

STOCHASTIC ANALYSIS AND R A N D O M M A P S IN HILBERT SPACE A.A. Dorogovtsev

///VSP///

Utrecht, The Netherlands, 1994

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

© VSP BV 1994 First published in 1994 ISBN 90-6764-163-4

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Dorogovtsev, A.A. Stochastic analysis and random maps in Hilbert space / A.A. Dorogovtsev. - Utrecht: VSP Withref. ISBN 90-6764-163-4 bound NUGI811 Subject headings: stochastic analysis / Hilbert space

Printed in The Netherlands by Koninklijke Wöhrmann bv, Zutphen.

Contents

Introduction 1

3

Stochastic calculus 1.1 Preliminaries (L2-theory) 1.2 Smooth open sets 1.3 The localization of the extended stochastic integral and the stochastic derivative 1.4 Stochastic integrals with respect to Gaussian random measures . . . . 1.5 Sobolev spaces on infinitely-dimensional domain

5 5 16 21 25 31

2 Random maps in Hilbert space 2.1 The definition of a random map 2.2 Gaussian strong random operators 2.3 The integral representation of random maps

37 37 38 46

3 The composition of random maps 3.1 The reasons 3.2 Random multi-linear Hilbert-Schmidt forms of the generalized Gaussian random elements 3.3 The action of random map on random elements 3.4 Bounded random operators 3.5 Generalized Gaussian functionals of the first kind 3.6 Generalized Gaussian functionals of the second kind and the Fourier transform

49 49

4 Stochastic analysis and quantum mechanics 4.1 Systems with changeable numbers of particles 4.2 The random statistics 4.3 Local intersection times

75

5 Equations with random operators 5.1 Estimation of the moments of multilinear forms from white noise . . .

83 83

1

49 54 58 63 67

75 78 80

2

CONTENTS 5.2 5.3 5.4

Equations with random operators under moment conditions Boundary value problems Stochastic calculus and partial differential equations of the first order

Bibliography

85 93 96 105

Introduction This book is devoted to stochastic operators in Hilbert space. A number of models in modern probability theory apply the notion of a stochastic operator in explicit or latent form. These models can be divided into two principal groups. T h e first group includes such objects as random matrices, random linear operators as defined by Bharycha-Reid, differential and integral equations with random coefficients and kernels etc. In models of this type a random operator is taken as the set of deterministic operators which are indicated by points from some probability space. In this case there is a "good" operator (matrix, differential or integral operator etc.) for every ui . T h e second group includes stochastic integrals with respect to random measures with independent values on disjoint sets and their generalizations as strong or weak random linear operators of Skorokhod. In this case there is no "good" deterministic operator for every u>, however the result of the action of the stochastic operator on elements of Hilbert space is observed. In this book, objects from t h e second group in the Gaussian case are considered. Therefore it is useful to consider all random variables and elements as functionals from t h e Wiener process or its formal derivative, i.e. white noise. T h e book consists of five chapters. T h e first chapter is devoted to stochastic calculus. Properties of t h e extended stochastic integral and stochastic derivative are considered. This introductory material is well-known and may be called L2theory because all random variables and elements to be considered have finite second moment. T h e main goal of t h e first chapter is to prepare t h e tools for solving stochastic equations. T h e condition of the existence of second moment is very strong in m a n y situations. So in this chapter stochastic calculus for random elements and variables which have no finite moments is built. T h e localization of t h e extended stochastic integral and stochastic derivative on a special set of random events is given. For certain generalized Wiener functionals, in the sense of Hida, ordinary random variables which coincide with them are found. Stochastic calculus of white noise is closely related to analysis in infinitelydimensional spaces. For this reason chapter 1 contains some statements a b o u t Sobolev spaces on open sets with Gaussian measure in Hilbert space. In t h e second chapter the structure of stochastic operators, mainly t h e s t r u c t u r e of Gaussian strong linear operators, is studied. For a certain class of such operators, i.e. those cases when t h e operator is a set of deterministic bounded operators indicated by the points of probability space, fi is described. For a general stochastic operator, the existence of representation as a sum of multiple stochastic integrals will be proved.

3

4

Introduction

In chapter 3 the definition of the action of stochastic operator on random elements is considered. As has been mentioned above, in particular cases the stochastic operator is the stochastic integral. So in these cases the stochastic integral for the random integrand must be defined. In this book the extended stochastic integral is used as a tool. Another method of approach to the definition of the action of a stochastic linear operator on random elements is based on the symmetric stochastic integral. The relationship of these two methods is considered. In particular a generalized Fourier transform is constructed for the generalized white noise functionals. This transform is used for the comparison of the extended stochastic integral with the symmetric stochastic integral. The localization from chapter 1 provides the possibility of determining the action of a random operator on a random element which has no finite moments. Chapter 4 is devoted to the mathematical models in which the notions of stochastic calculus arise. Local time intersection for the families of Markov processes, random statistics and systems with random values of particles is considered. In chapter 5 the equation with random operators is considered. Theorems of the existence and uniqueness of solutions and the smoothness of solution in particular cases are proved. Finally, I'd like to express my gratitude to academicians Anatoly Vladimirovich Skorokhod and Yury Lvovich Daletsky for useful discussions and questions.

Chapter 1 Stochastic calculus 1.1

Preliminaries (L 2 -theory)

This section is devoted to the definitions of the extended stochastic integral and stochastic derivative and to their properties in the case when all random elements and variables, which are considered, have finite second moment. Nowadays there are many excellent articles devoted to this subject. They are mentioned in the reference list. He who wants to study stochastic calculus in detail, can read the pioneering works of M. Hitsuda, A. V. Skorokhod and Yu. L. Daletsky and N. V. Paramonova or such surveys as the articles of T. Sekiguchy and Y. Shiota, E. Pardoux and D. Nualart. This section contains well-known results and new facts which will be useful later. The statements are formulated in a convenient manner in order to provide the following exposition. Let ( f t , J7, P) be a probability space and H be a real separable Hilbert space with inner product (• •) and norm || • ||. Let £ denote the generalized Gaussian random element in H which has zero mean and identical correlation operator. In other words £ is a linear map which maps elements of H into the set of Gaussian random variables and has the following property: V *>€//:

E((, 0 are n-dimensional Hermite polynomials. E x a m p l e 1.1.2'. In this situation, for each k > 1, space Hk is a symmetric part of x . . . x X , 1. E x a m p l e 1.1.3'. Space B* is dense in H. Hence for every Qk € Hk, k > 1, the random variable Qk(£, •••,£) is defined by the function Qk : B —• R which is a ¿/-limit of the sequence of usual polynomials of i - t h degree on B. S o Qk is a measurable polynomial with respect to measure u. Now consider the new important notion — stochastic derivative. Begin from the situation of example 1.1.3. T h e usual definition of the derivative for t h e function / : B —• R is based on consideration of the ratio fte{v,h)

=

—

,

u>,h£B.

If t h e function / is d i f f e r e n t i a t e at every point of the space B , then B :

lim U,(u,h) At—* 0

=

(f(u),h),

Preliminaries

9

(L^-theory)

where / ' is the derivative of / . Hence, for the random variable / , there exists v — lim /(•, h). v

At—»0

'

This fact provides the possibility of defining the derivative of a random variable, but there are two difficulties using this method. Firstly, the random variable /a 0 : E(Qt,At(", h) - kQk{h, Therefore, for multiple forms from

0)2

0,

the following definition is obtained.

Definition 1 . 1 . 1 . Stochastic derivative of Qk(£, •••,0 random element DQk(£, . . . , £ ) in H such that VAefl:

Ai —» 0.

(DQk(t,...,£),h)

= kQk{h,t,...,t)

for Qk G Hk, k > 1 is a

(mod P).

This definition is based on the usual definition of the derivative, but does not depend on the situation of example 1.1.3. The stochastic derivative for the random variable 7 € L ^ i l , ^ , P) can now be defined with the help of the Ito-Wiener expansion. Definition 1 . 1 . 2 . The random variable 7/ € ¿ 2 ( 0 , P) with the Ito-Wiener expansion (1.1.1) is stochastically differentiate if the series 00 (1.1.2)

k=0

converges on the square mean in H. The sum of (1.1.2) is called a stochastic derivative of 7/ and is denoted by Dt). Before studying the properties of a stochastic derivative, D, consider some examples. E x a m p l e 1 . 1 . 4 . Consider H = R. Then £ is a usual Gaussian random variable with a zero mean and variance 1. Multiple Hilbert-Schmidt forms from £ are Hermite polynomials, and V k > 0 :

DHk(0

= H'k(0

=

kHk-,(£),

where the symbol " ' " denotes a usual derivative. Therefore, the random variable a = /(£) € is stochastically differentiate if, and only if, / has a

CHAPTER

10

1. Stochastic calculus

derivative almost everywhere on R , is absolutely continuous and /'(£) 6 ¿2 ( f i , T , Under this condition D f ( £ ) = / ' ( £ ) .

P).

E x a m p l e 1.1.5. Case H = RJ1. Now £ = where are independent identically distributed Gaussian random variables with zero mean and variance 1. Multiple Hilbert-Schmidt forms from £ are multi-dimensional Hermite polynomials from £ 1 , . . . , £n and, as in example 1.1.4, a stochastic derivative for them is a usual gradient. For example, if n = 2, k = 2, and form Q? is determined by matrix

*=(!i) in the natural basis ei = (1; 0), e 2 = (0; 1), then

D Q i i t u h ) = 2 Q 2 ( . , 0 = 2 6 e2 +

= VQ(6,6),

where