Table Of ContentDiffusions and Waves
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 552
Diffusions and Waves
by
Henryk Gzyl
SPRINGER -SCIENCE+BUSINESS MEDIA, B.V.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-3949-9 ISBN 978-94-010-0293-6 (eBook)
DOI 10.1007/978-94-010-0293-6
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© 2002 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2002
Softcover reprint of the hardcover 1s t edition 2002
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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CONTENTS
Introduction 1
Chapter 1. Basic Probabilistic Notions 3
1.1. Probability spaces 3
1.2. Conditional expectations 6
1.3. Examples of conditional expectations 7
1.4. Processes 9
1.5. Semi groups, generators and resolvents 12
1.5.1. Semigroups 12
1.5.2. Resolvents 14
1.6. Examples 15
1.7. Construction of Markov processes 17
1.7.1. Basics 17
1.7.2. Sketch of constructions 19
1.7.3. Stopping times and the strong Markov property 20
1.8. Transformations of Markov processes 23
I.B.I. Additive and multiplicative functionals 23
I.B.2. Time changing a Markov process 26
I.B.3. The killing of a Markov process 27
References 29
Chapter 2. From Brownian Motion to Diffusions 31
2.1. Brownian motion 31
2.1.1. The classical approach 31
2.1.2. The modern approach 34
2.1.3. Some important hitting times and hitting distributions 34
2.1.4. Dirichlet problem for the Laplacian 37
2.2. Diffusions 39
v
vi Contents
2.2.1. Stochastic integration with respect to Brownian motion 40
2.2.2. The Cameron-Martin-Girsanov-Maruyama transformation 45
2.3. Diffusions as solutions of stochastic equations 46
2.3.1. The basic stochastic differential equation 47
2.3.2. The Markov nature of the solutions 48
2.4. Reflected diffusions 49
2.5. Killed diffusions and some fundamental identities 53
References 57
Chapter 3. Waves 59
Introduction 59
3.1. Waves of constant speed in ~d 60
3.1.1. Waves in one-dimensional space 61
3.1.2. Waves in spaces of dimension d > 1: Averaging methods 62
3.1.3. Waves in spaces of dimension d > 1. The Fourier transform approach 65
3.1.4. Fourier transform in time 65
3.1.5. Fourier transform in space 67
3.1.6. Expansion of the solution in plane waves 68
3.1.7. The (de)complexification approach 68
3.1.8. Factorization 69
3.2. Propagators and Green functions 72
3.3. Geometrical optics 77
3.3.1. Basics 77
3.3.2. Solving the eikonal equation 78
3.3.3. Solving the transport equations 81
3.4. General representation of solutions 83
References 87
Chapter 4. Waves and Brownian Motions 89
4.1. Waves in full space 89
4.2. Dirichlet problems 92
4.2.1. Waves in half space with Dirichlet boundary conditions 93
4.2.2. Waves in a sphere with Dirichlet boundary conditions 95
4.3. Neumann type boundary conditions 96
4.4. Existence results 98
4.5. Problems of Dirichlet type in unbounded domains: from the Markov property
to the Huygens condition and the Sommerfeld radiation condition 99
4.5.1. The Dirichlet problem in unbounded domains 99
4.5.2. From the Markov property to the Huygens construction 100
4.5.3. Sommerfeld's radiation condition 101
4.6. Extended Hadamard's construction 104
4.7. From resolvents to propagators 105
4.8. Reciprocity: A probabilistic approach 107
References 110
Contents vii
Chapter 5. Waves and Diffusions 113
5.1. Waves in full space 113
5.2. Existence of solutions to the wave equations 116
5.3. An evaluation of some path integrals 118
SA. Waves in stratified media 119
5.4.1. Waves in a one-dimensional stratified medium 120
5.4.2. Waves in a stratified half-space 123
5.4.3. A change of spatial scales 124
5.4.4. Waves in multidimensional stratified media 126
5.4.5. Examples 127
5.5. Maxentropic equivalent linearization and approximate solutions to the wave
equations 130
References 133
Chapter 6. Asymptotic Expansions 135
6.1. Digressive introduction 135
6.2. Probabilistic approach to geometrical optics 141
6.3. Geometrical optics and the Dirichlet problem 143
6A. Two variations on a theme 145
6.4.1. The first variation on the theme 145
6.4.2. The second variation on the theme 147
6.5. Geometrical optics and the Neumann problem 148
6.6. Example 149
6.7. Long time asymptotics 152
References 154
Chapter 7. Transmutation Operations 155
7.1. Basic transmutations 155
7.2. Probabilistic version of transmutation operations 158
7.3. Examples 162
7 A. More inversion techniques and simple examples 165
7.5. The ascent method 170
7.6. The closing of the circle. Some heuristics 174
References 175
Chapter 8. More Connections 177
8.1. Waves in discrete structures and Markov chains 177
8.2. Approximate Laplacians, regular jump processes and random flights 180
8.2.1. Approximate Laplacians 180
8.3. Regular jump processes 182
8A. Random flights 185
8.5. Random evolutions 187
8.6. First-order hyperbolic systems 192
viii Contents
8.6.1. One-dimensional stratified media a la Gaveau 194
8.6.2. Thomas' completion of Feynmann's suggestion 195
8.7. Pseudo processes and Euler's equation 198
8.8. Damped waves: Playing with a simple model 201
References 206
Chapter 9. Applications 209
9.1. An inverse source problem 209
9.1.1. A basic result 209
9.1.2. Equivalent source problem 211
9.1.3. Solving the integral geometry problem 213
9.1.4. Some very simple examples 215
9.2. Probabilistic approach to a discrete inverse problem 217
9.3. Dependence of boundary data on propagation velocity 220
9.4. The Born approximation 221
9.4.1. Inversion of the Born approximation 224
9.5. Scattering by a bounded object 228
9.5.1. Point sources 231
References 234
Subject Index 235
INTRODUCTION
Probability theory, and more specifically, probabilistic diffusion theory or functional in
tegration has played a significant role in the theory of elliptic and parabolic differential
equations. Take a look at Sections 1.1 and 2.1 to get some of the flavor of the field.
The basic link is provided by the probabilistic representation of the time-dependent
Green functions (or propagators in the physicists' jargon) and the time-independent Green
functions as path integrals. This connection between partial differential equations and dif
fusions does two things for us. On the more theoretical side, it provides us with elegant
proofs of very many results, and, on the practical side, it provides us with explicit formu
lae which can be used (and have been used) as starting points for numerical work. As a
matter of fact, the computations of Green functions from their probabilistic representation
constitute outstanding examples of trivially parallelizable computations: at each CPU one
averages over all trajectories issued from a given discretization point.
Actually, the material in this book was developed as a response to a challenge by some
geophysicists friends of mine: How about putting all that mathematics to use to show us
what a wave in a heterogeneous medium looks like!
So the first question was: How to associate a diffusion process with a wave in a hetero
geneous medium? Or even simpler: How to associate a diffusion process with a wave
propagating in a constant velocity medium? This problem seems to be open despite the
fact that a negative answer provided by Duddley in [12-13] of Chapter 8 exists. If the
wave equation in a three-dimensional space is Lorentz invariant, and there are no Lorentz
invariant diffusions, we seem to be in a dead-end street. On the other hand, if the wave
equation is factored into Dirac equations (or some other procedure of extracting square
roots) and a way of associating processes with these equations exists, there may exist a
way of going around Duddley's negative result. This is the spirit in which the reader should
look at the results in Chapters 7 and 8: A quest to directly associate stochastic processes
with wave equations.
Chapters 1, 2 and 3 are intended to make the volume self-contained, and there is a long
list of commented definitions and the very basic stuff. Chapters 4 and 5 contain the bulk
of the material on the best I can do so far in connecting waves to diffusions, first for the
H. Gzyl, Diffusions and Waves
© Kluwer Academic Publishers 2002
2 Introduction
constant velocity case and then for scalar waves in heterogeneous media. The connection
comes about by reduction to the elliptic case by means of Laplace transformation in the
time variable.
The Laplace transform is used for two reasons: first because it is the natural way to do
away with the time variable when dealing with problems with initial data in time. But this
is not the important reason, for initial data can be made part of the source terms driving the
wave. The important reason is that by taking Laplace transforms, instead of the traditional
Fourier transforms used in scattering theory, the resulting elliptic problem is amenable to
the direct and traditional probabilistic analysis. Perhaps our colleagues working with path
integration in quantum mechanics could provide analogous results for the elliptic problems
obtained by taking Fourier transforms.
As we shall see in Chapters 4, 5, 6, and 9, the probabilistic representation formulae
will be important, not only in obtaining the solution (explicit or implicit) of initial and
boundary value problems. It will be useful as well as a starting point for the solution of
inverse problems.
This, as will be clearly seen below, is due to the fact that in the path integral or proba
bilistic, representations of the propagators and static Green functions, the velocity of prop
agation appears in a very explicit way.
The results collected in this volume are a first shot at very interesting inverse problems
related to wave propagation. Many interesting open problems exist and my hope is that this
volume may be of help and motivate others to look at them (from the present perspective,
of course).
