Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach

by Holden, Helge; Oksendal, Bernt; Uboe, Jan; Zhang, Tusheng

• ISBN13: 9780387894874
• ISBN10: 038789487X
• Edition: 2nd
• Format: Paperback
• Copyright: 2009-12-30
• Publisher: Springer Verlag

Summary

The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs. In this, the second edition, the authors build on the theory of SPDEs driven by space-time Brownian motion, or more generally, space-time Lévy process noise, and introduce new applications of the field.Because the authors allow the noise to be in both space and time, the solutions to SPDEs are usually of the distribution type, rather than a classical random field. To make this study rigorous and as general as possible, the discussion of SPDEs is therefore placed in the context of Hida white noise theory. The key connection between white noise theory and SPDEs is that integration with respect to Brownian random fields can be expressed as integration with respect to the Lebesgue measure of the Wick product of the integrand with Brownian white noise, and similarly with Lévy process. The first part of the book deals with the classical Brownian motion case. The second extends it to the Lévy white noise case. For SPDEs of the Wick type, a general solution method is given by means of the Hermite transform, which turns a given SPDE into a parameterized family of deterministic PDEs. Applications of this theory are emphasized throughout. The stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance.Graduate students in pure and applied mathematics as well as researchers in SPDEs, physics, and engineering will find this introduction indispensible. Useful exercises are collected at the end of each chapter.From the reviews:"The authors have made significant contributions to each of the areas. As a whole, the book is well organized and very carefully written and the details of the proofs are basically spelled out... This is a rich and demanding book'¦ It will be of great value for students of probability theory or SPDEs with an interest in the subject, and also for professional probabilists."American Mathematical Society 1996"...a comprehensive introduction to stochastic partial differential equations."Zentralblatt 1996

Table of Contents

Preface to the Second Edition	p. ix
Preface to the First Edition	p. xi
Introduction	p. 1
Modeling by Stochastic Differential Equations	p. 1
Framework	p. 13
White Noise	p. 13
The 1-Dimensional, d-Parameter Smoothed White Noise	p. 13
The (Smoothed) White Noise Vector	p. 20
The Wiener-Itô Chaos Expansion	p. 21
Chaos Expansion in Terms of Hermite Polynomials	p. 21
Chaos Expansion in Terms of Multiple Itô Integrals	p. 29
The Hida Stochastic Test Functions and Stochastic Distributions. The Kondratiev Spaces (S)m;N, (S)m;N-?	p. 31
The Hida Test Function Space (S) and the Hida Distribution Space (S)*	p. 40
Singular White Noise	p. 42
The Wick Product	p. 43
Some Examples and Counterexamples	p. 47
Wick Multiplication and Hitsuda/Skorohod Integration	p. 50
The Hermite Transform	p. 61
The (S)N?,r Spaces and the S-Transform	p. 75
The Topology of (S)N-1	p. 81
The F-Transform and the Wick Product on L1(&mu)	p. 88
The Wick Product and Translation	p. 92
Positivity	p. 98
Applications to Stochastic Ordinary Differential Equations	p. 115
Linear Equations	p. 115
Linear 1-Dimensional Equations	p. 115
Some Remarks on Numerical Simulations	p. 118
Some Linear Multidimensional Equations	p. 119
A Model for Population Growth in a Crowded, Stochastic Environment	p. 120
The General (S)-1 Solution	p. 121
A Solution in L1(&mu)	p. 123
A Comparison of Model A and Model B	p. 127
A General Existence and Uniqueness Theorem	p. 128
The Stochastic Volterra Equation	p. 131
Wick Products Versus Ordinary Products: a Comparison Experiment	p. 140
Variance Properties	p. 143
Solution and Wick Approximation of Quasilinear SDE	p. 145
Using White Noise Analysis to Solve General Nonlinear SDEs	p. 150
Stochastic Partial Differential Equations Driven by Brownian White Noise	p. 159
General Remarks	p. 159
The Stochastic Poisson Equation	p. 161
The Functional Process Approach	p. 163
The Stochastic Transport Equation	p. 164
Pollution in a Turbulent Medium	p. 164
The Heat Equation with a Stochastic Potential	p. 169
The Stochastic Schrödinger Equation	p. 169
L1(&mu)Properties of the Solution	p. 172
The Viscous Burgers Equation with a Stochastic Source	p. 178
The Stochastic Pressure Equation	p. 186
The Smoothed Positive Noise Case	p. 187
An Inductive Approximation Procedure	p. 192
The 1-Dimensional Case	p. 193
The Singular Positive Noise Case	p. 194
The Heat Equation in a Stochastic, Anisotropic Medium	p. 195
A Class of Quasilinear Parabolic SPDEs	p. 200
SPDEs Driven by Poissonian Noise	p. 203
Stochastic Partial Differential Equations Driven by Lévy Processes	p. 213
Introduction	p. 213
The White Noise Probability Space of a Lévy Process (d = 1)	p. 215
White Noise Theory for a Lévy Process (d = 1)	p. 219
Chaos Expansion Theorems	p. 219
The Lévy-Hida-Kondratiev Spaces	p. 225
White Noise Theory for a Lévy Field (d ≥ l)	p. 232
Construction of the Lévy Field	p. 232
Chaos Expansions and Skorohod Integrals (d ≥ 1)	p. 238
The Wick Product	p. 244
The Hermite Transform	p. 246
The Stochastic Poisson Equation	p. 248
Waves in a Region with a Lévy White Noise Force	p. 252
Heat Propagation in a Domain with a Lévy White Noise Potential	p. 253
Appendix A	p. 257
Appendix B	p. 263
Appendix C	p. 271
Appendix D	p. 273
Appendix E	p. 281
References	p. 289
List of frequently used notation and symbols	p. 297
Index	p. 303
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