Table Of ContentIntroduction to Infinite Dimensional Stochastic Analysis
Mathematics and Its Applications
Managing Editor
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 502
Introduction to Infinite
Dimensional Stochastic
Analysis
By
Zhi-yuan Huang
Department of Mathematics,
Huazhong University ofScience and Technology,
Wuhan P. R. China
and
Jia-an Yan
Institute of Applied Mathematics,
Chinese Academy of Sciences,
Beijing P. R. China.
Science Press
Beijing/New York,
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-5798-1 ISBN 978-94-011-4108-6 (eBook)
DOI 10.1007/978-94-011-4108-6
This is an updated and revised translation of the original Chinese publicat ion
©Science Press, Beijing, P. R. China, 1997.
Printed an acid-free paper
AII Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000
Softcover reprint ofthe hardcover lst edition 2000
N o 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 informat ion storage and
retrieval system, without written permission from the copyright owners.
Contents
Preface ................................................................... 1X
Chapter I Foundations of Infinite Dimensional Analysis ........... 1
§1. Linear operators on Hilbert spaces ........................... 1
1.1 Basic notions, notations and lemmas ............................. 1
1.2 Closable, symmetric and self-adjoint operators ................... 4
1.3 Self-adjoint extension of a symmetric bounded below operator ... 8
1.4 Spectral resolution of self-adjoint operators ..................... 10
1.5 Hilbert-Schmidt and trace class operators ...................... 14
§2. Fock spaces and second quantization ....................... 19
2.1 Tensor products of Hilbert spaces .............................. 19
2.2 Fock spaces .................................................... 24
2.3 Second quantization of operators ............................... 26
§3. Countably normed spaces and nuclear spaces .............. 29
3.1 Countably normed spaces and their dual spaces ................. 30
3.2 Nuclear spaces and their dual spaces ........................... 34
3.3 Topological tensor product, the Schwartz kernels theorem ....... 38
§4. Borel measures on topological linear spaces ............... .41
4.1 Minlos-Sazanov theorem ........................................ 41
4.2 Gaussian measures on Hilbert spaces ........................... 48
4.3 Gaussian measures on Banach spaces ........................... 51
Chapter II Malliavin Calculus ...................................... 59
§1. Gaussian probability spaces and Wiener chaos
decomposition ................................................. 59
1.1 Functionals on Gaussian probability spaces ..................... 59
1.2 Numerical models .............................................. 64
1.3 Multiple Wiener-Ito integral representation ..................... 67
§2. Differential calculus of functionals, gradient
and divergence operators .................................... 72
2.1 Finite dimensional Gaussian probability spaces ................. 72
2.2 Gradient and divergence of smooth functionals ................. 76
2.3 Sobolev spaces of functionals ................................... 81
§3. Meyer's inequalities and some consequences ............... 86
3.1 Ornstein-Uhlenbeck semigroup ................................. 86
VI Contents
3.2 LP-multiplier theorem .......................................... 89
3.3 Meyer's inequalities ............................................ 92
3.4 Meyer-Watanabe's generalized functionals ...................... 97
§4. Densities of non-degenerate functionals ................... 100
4.1 Malliavin covariance matrices, some lemmas ................... 101
4.2 Existence of densities .......................................... 103
4.3 Smoothness of densities ....................................... 106
4.4 Examples ..................................................... 110
Chapter III Stochastic Calculus of Variation for
Wiener Functionals ........................................ 113
§1. Differential calculus of Ito functionals and
regularity of heat kernels ................................... 113
1.1 Skorohod integrals ............................................ 113
1.2 Smoothness of solutions to stochastic differential equations .... 118
1.3 Hypoellipticity and Hormander's conditions ................... 120
1.4 A probabilistic proof of Hormander's theorem ................. 125
§2. Potential theory over Wiener spaces and
quasi-sure analysis ........................................... 130
2.1 (k,p)-capacities ................................................. 130
2.2 Quasi-continuous modifications ................................ 133
2.3 Tightness, continuity and invariance of capacities .............. 135
2.4 Positive generalized functionals and measures with
finite energy .................................................... 139
2.5 Some quasi-sure sample properties of stochastic processes ...... 142
§3. Anticipating stochastic calculus ............................ 145
3.1 Approximation of Skorohod integrals by Riemannian sums ..... 145
3.2 Ito formula for anticipating processes ......................... 149
3.3 Anticipating stochastic differential equations .................. 155
Chapter IV General Theory of White Noise Analysis ........... 161
§1. General framework for white noise analysis ............... 162
1;1 Wick tensor products and the Wiener-Ito-Segal isomorphism .. 162
1.2 Testing functional space and distribution space ................ 165
1.3 Classical framework for white noise analysis ................... 169
§2. Characterization of functional spaces ...................... 171
2.1 s-transform and characterization of space (E)~/(O::;,8<I) ........ 171
(Ere
2.2 Local s-transform and characterization of space 1 •••••••••• 177
2.3 Two characterizations for testing functional spaces ............ 179
2.4 Some examples of distributions ................................ 183
Contents vii
§3. Products and Wick products of functionals ............... 188
3.1 Products of functionals ........................................ 188
3.2 Wick products of distributions ................................ 191
3.3 Application to Feynman integrals ............................. 193
§4. Moment characterization of distributions and
positive distributions ........................................ 195
4.1 The renormalization operator ................................. 195
4.2 Moment characterization of distribution spaces ................ 197
4.3 Measure representation of positive distributions ............... 199
4.4 Application to p(4)).-quantum fields ........................... 206
Chapter V Linear Operators on Distribution Spaces ............ 210
§1. Analytic calculus for distributions ......................... 210
1.1 Scaling transformations ....................................... 210
1.2 Shift operators and Sobolev differentiations .................... 212
1.3 Gradient and divergence operators ............................ 216
§2. Continuous linear operators on distribution
spaces ......................................................... 219
2.1 Symbols and chaos decompositions for operators ............... 219
2.2 s-transforms and Wick products of generalized operators ...... 224
§3. Integral kernel operators and integral kernel
representation for operators ................................ 229
3.1 Contraction of tensor products ................................ 229
3.2 Integral kernel operators ...................................... 231
3.3 Integral kernel representation for generalized operators ........ 237
§4. Applications to quantum physics ........................... 240
4.1 Quantum stochastic integrals .................................. 240
4.2 Klein-Gordon field ............................................ 243
4.3 Infinite dimensional classical Dirichlet forms ................... 245
Appendix A Hermite polynomials and Hermite functions ........ 252
Appendix B Locally convex spaces and their dual spaces ........ 257
1. Semi-norms, norms and H-norms ............................... 257
2. Locally convex topological linear spaces, bounded sets .......... 258
3. Projective topologies and projective limits ...................... 259
4. Inductive topologies and inductive limits ....................... 260
5. Dual spaces and weak topologies ............................... 261
6. Compatibility and Mackey topology ............................ 262
7. Strong topologies and reflexivity ................................ 263
VIll Contents
8. Dual maps ..................................................... 263
9. Uniformly convex spaces and Banach-Saks' theorem ............ 264
Comments .............................................................. 266
References .............................................................. 271
Subject Index .......................................................... 290
Index of Symbols ...................................................... 294
Preface
The infinite dimensional analysis as a branch of mathematical sciences was
formed in the late 19th and early 20th centuries. Motivated by problems in
mathematical physics, the first steps in this field were taken by V. Volterra, R.
GateallX, P. Levy and M. Frechet, among others (see the preface to Levy[2]).
Nevertheless, the most fruitful direction in this field is the infinite dimensional
integration theory initiated by N. Wiener and A. N. Kolmogorov which is closely
related to the developments of the theory of stochastic processes. It was Wiener
who constructed for the first time in 1923 a probability measure on the space of
all continuous functions (i.e. the Wiener measure) which provided an ideal math
ematical model for Brownian motion. Then some important properties of Wiener
integrals, especially the quasi-invariance of Gaussian measures, were discovered
by R. Cameron and W. Martin[l, 2, 3]. In 1931, Kolmogorov[l] deduced a second
order partial differential equation for transition probabilities of Markov processes
with continuous trajectories (i.e. diffusion processes) and thus revealed the deep
connection between theories of differential equations and stochastic processes.
The stochastic analysis created by K. Ito (also independently by Gihman [1])
in the forties is essentially an infinitesimal analysis for trajectories of stochastic
processes. By virtue of Ito's stochastic differential equations one can construct
diffusion processes via direct probabilistic methods and treat them as function
als of Brownian paths (i.e. the Wiener functionals). This affords a possibility
of using probabilistic methods to investigate deterministic differential equations
and many other pure analytical problems. On the other hand, during the same
decade of this century, the famous work on functional integration approach to
mathematical physics derived by R. Feynman and M. Kac as well as rapid devel
opments in quantum field theory gave new impulsions to the analysis in infinite
dimension.
The classical notions of functions and derivatives in finite dimensional analy
sis were long felt to be restrictive in mathematical physics. Notions of generalized
functions and derivatives were first introduced by S. L. Sobolev[l] in 1936 to meet
the needs of solving equations in mathematical physics. The theory of Sobolev
spaces has played an important role in modern treatment of partial differential
operators. L. Schwartz systematically developed this idea and established theory
x Preface
of distributions. Many singular objects in classical physics such as Dirac delta
functions thus obtained mathematically rigorous meanings. However, up to now
there still exist a lot of intuitive notions and heuristic calculations in physics
which remain meaningless from the mathematical viewpoint. To put them on a
sound mathematical foundation is quite important for the development of theo
retical physics and is a real challenge to mathematicians and physicists.
The same situation also occurred in infinite dimensional analysis. Since many
important functionals (e.g. diffusion processes regarded as Wiener functionals)
are not differentiable in Fh~chet sense, it is essential to generalize the notions of
functionals and differentiation in infinite dimensional spaces. In 1976, P. Malli
avin[l] successfully extended the gradient, divergence and Ornstein-Uhlenbeck
operators to infinite dimensional cases and created the stochastic calculus of
variation (known as Malliavin calculus). Under his sense of differentiation, many
important Wiener functionals become smooth (infinitely differentiable). Along
this line S. Watanabe[lJ, I. Shigekawa[lJ, D. W. Stroock[lJ, P. A. Meyer[l] et
al. established a Sobolev theory over infinite dimensional spaces. With fruitful
applications to partial differential operators and heat kernels, to stochastic oscil
latory integrals, to filtering and control of stochastic systems, etc., the Malliavin
calculus has become one of the most significant successes in the field of stochastic
analysis.
In 1975, T. Hida launched out the white noise analysis. Since Gaussian
white noise is the derivative of Brownian motion in the distribution sense, its
sample space lies in the space of Schwartz distributions. By regarding Wiener
functionals as functionals of white noise, Hida established a Schwartz theory over
infinite dimensional spaces. With profound background in physics and successful
applications to Feynman integrals as well as quantum field theory, white noise
analysis has attracted more and more attention from theoretical physicists.
These two frameworks of infinite dimensional analysis are essentially based
on the quasi-invariance of Gaussian measures and could be unified into one set
ting of the so-called Gaussian probability spaces. Its origin goes back to the
works on abstract integration on Hilbert spaces by I. E. Segal[l, 2] and on rigged
Hilbert spaces by I. M. Gel'fand in the fifties (see Gel'fand & Vilenkin[l]), and
the work on abstract Wiener spaces by L. Gross[l] in the sixties. The choice of
frameworks depends naturally on the practical problems to be solved. In Malli
avin calculus, for example, one requires the space of testing functionals to be
rich enough so that many important functionals become smooth, while in Hida
calculus, one hopes that the space of distributions would be sufficiently large to
contain many singular objects in physics which are not rigorously defined so far.
The relationship between these two kinds of calculus is quite similar to that of
Sobolev and Schwartz theories in finite dimensions.
This book is intended to offer a quick introduction to the above mentioned
rapidly developing research area --infinite dimensional stochastic analysis. We
