White Noise: An Infinite Dimensional Calculus

528 Pages·1993·12.943 MB·English

by  Takeyuki Hida, Hui-Hsiung Kuo, Jürgen Potthoff, Ludwig Streit (auth.)

White Noise
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 253
White Noise
An Infinite Dimensional Calculus
by
Takeyuki Hida
Department ofM athematics,
Meijo University,
Nagoya, Japan
Hui-Hsiung Kuo
Department of Mathematics,
Louisiana State University,
Baton Rouge, US.A.
Jtirgen Potthoff
Lehrstuhl./Ur Mathematik V,
Universitiit Mannheim,
Mannheim, Germany
and
Ludwig Streit
BiBos, Universitiit Bielefeld,
Bielefeld, Germany, and
Universidade da Madeira, Funchal, Madeira
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4260-6 ISBN 978-94-017-3680-0 (eBook)
DOI 10.1007/978-94-017-3680-0
Printed on acid-free paper
AII Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
Softcover reprint of the hardcover 1s t edition 1993
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.
To
Minami, Fukuko, Tomiko, Clara and Jan
Contents
1. GauBBian Spaces ................................................................................................ 1
2. <J' and ':I Transformation and the Decomposition Theorem ............................... 10
3. Generalized Functionals ................................................................................... 35
3.A The Spaces (L2)± ......................................................................................... 36
3.B Triples of Functional Spaces .......................................................................... 52
3.C Constructions with Second Quantized Operators ........................................... 61
4. The Spaces (':I) and (':1)* ........................................•......................................... 74
4.A Basic Properties and Examples ..................................................................... 74
4.B Positive Generalized Functionals ................................................................ 109
4.C Characterization Theorems and Applications ............................................... 119
4.D Analytic Version o£ (1) and Growth Estimates .......................................... 136
5. Calculus of Differential Operators .................................................................. 146
5.A Pointwise Calculus of Differential Operators ............................................... 147
5.B Differential Operators on (1) ....................................................................... 154
5.C Differential Operators and Their Adjoints on (L2) and (1)* ........................ 163
5.D Table of Rules ............................................................................................ 179
6. Laplacian Operators ...................................................................................... 184
6.A White Noise Kernel Operators ..................................................................... 184
6.B The Number Operator ................................................................................. 196
6.C The Gross and Volterra Laplacians ............................................................. 199
6.D The Levy Laplacian .................................................................................... 211
6.E The Commutation Relations ....................................................................... 228
viii Contents
7. The Spaces '!J) and '!J)* •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 232
7.A The Ornstein-Uhlenbeck Semigroup ........................................................... 233
7.B Littlewood-Paley-Stein Inequalities ............................................................ 249
7.C Meyer's Inequalities and Their Applications ................................................ 263
8. Stochastic Integration .................................................................................... 277
8.A Introduction ................................................................................................ 277
8.B Integrals of Ito Type ................................................................................... 283
8.C Stochastic Integrals with Anticipating Integrands ........................................ 289
8.D Properties of Anticipating Stochastic Integrals ............................................ 300
9. Fourier and Fourier-Mehler Transforms ......................................................... 317
9.A A Heuristic Motivation ............................................................................... 318
9.B Definition and Examples of the Fourier Transform ...................................... 319
9.C Elementary Properties ................................................................................. 324
9.D The Fourier Transform as an Adjoint Operator ........................................... 332
9.E A Characterization of the Fourier Transform ............................................... 336
9.F Relations with Laplacians, Wick Products, and Convolutions ...................... 339
9.G Relation with the Finite Dimensional Fourier Transform ............................ 348
9.H Fourier-Mehler Transforms ........................................................................ 356
10. Dirichlet Forms ........................................................................................... 366
10.A Construction of Energy Forms over !fl (lR) .................................................. 367
10.B Closability, and the Associated Diffusion Processes .................................... 382
11. Applications to Quantum Field Theory ........................................................ 399
11.A Introduction .............................................................................................. 399
l1.B Free Fields ................................................................................................ 405
l1.C Interacting Fields and the Role of <p - Bounds .......................................... 418
Contents ix
12. Feynman Integrals ....................................................................................... 435
12.A Construction of the Feynman Integrand: Free Particles ............................ 436
12.B Construction of the Feynman Integrand: Interacting Particles .................. .439
12.C Manipulating the Feynman Integrand: Schwinger's Variational Principle .. .448
Appendices ........................................................................................................ 451
A.l Hermite Polynomials .................................................................................. 451
A.2 Fock Space ................................................................................................. 456
A.3 Reproducing Kernel Hilbert Spaces ............................................................. 468
A.4 Frechet and Gateaux Derivatives on !f(lRd) •..•..•..••.•..•.•.....•.•.•...••••.•.•.•.•..•••. 472
A.5 Topologies on !f(lRd) and !fl (IR) .•..•...•.••.•...•.•...••.......•.••...•.•..•..•.•.•.....••..•...•. 477
Bibliography ..................................................................................................... 486
Notations and Conventions ................................................................................ 507
Index ................................................................................................................. 513
Preface
Many areas of applied mathematics call for an efficient calculus in infinite
dimensions. This is most apparent in quantum physics and in all disciplines of
science which describe natural phenomena by equations involving stochasticity.
With this monograph we intend to provide a framework for analysis in infinite
dimensions which is flexible enough to be applicable in many areas, and which on
the other hand is intuitive and efficient. Whether or not we achieved our aim must
be left to the judgment of the reader.
This book treats the theory and applications of analysis and functional
analysis in infinite dimensions based on white noise. By white noise we mean the
generalized Gaussian process which is (informally) given by the time derivative of
the Wiener process, i.e., by the velocity of Brownian mdtion. Therefore, in essence
we present analysis on a Gaussian space, and applications to various areas of
sClence.
Calculus, analysis, and functional analysis in infinite dimensions (or
dimension-free formulations of these parts of classical mathematics) have a long
history. Early examples can be found in the works of Dirichlet, Euler, Hamilton,
Lagrange, and Riemann on variational problems. At the beginning of this century,
Frechet, Gateaux and Volterra made essential contributions to the calculus of
functions over infinite dimensional spaces. The important and inspiring work of
Wiener and Levy followed during the first half of this century. Moreover, the
articles and books of Wiener and Levy had a view towards probability theory.
Almost from its very beginning quantum physics has been using infinite
dimensional calculus and it became an important (though sometimes informal)
tool in modern theoretical and mathematical physics through the pioneering work
of Berezin, Cook, Feynman, Fock, Schwinger, and Segal. During the sixties and
seventies, in the context of Euclidian and constructive quantum field theory, many
structures and deep results in infinite dimensional analysis have been worked out
by Albeverio and H~egh-Krohn, Glimm and Jaffe, Gross, Nelson, Simon, and
xi