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

✍ Scribed by Holden, Helge;Øksendal, Bernt;Uboe, Jan;Zhang, Tusheng

• Publisher: Springer
• Year: 2009
• Tongue: English
• Leaves: 311
• Series: Probability and Its Applications
• Category: Library

✦ Synopsis

The first edition ofStochastic 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 Levy process noise. Applications of the 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.

✦ Table of Contents

Preface to the Second Edition......Page 0
Preface to the First Edition......Page 8
Contents......Page 10
1.1 Modeling by Stochastic Differential Equations......Page 13
2.1.1 The 1-Dimensional, d-Parameter Smoothed White Noise......Page 24
2.1.2 The (Smoothed) White Noise Vector......Page 31
2.2.1 Chaos Expansion in Terms of Hermite Polynomials......Page 32
2.2.2 Chaos Expansion in Terms of Multiple Itô Integrals......Page 40
2.3 The Hida Stochastic Test Functions and StochasticDistributions. The Kondratiev Spaces (S)m;N,(S)-m;N......Page 42
2.3.1 The Hida Test Function Space (S) and the Hida Distribution Space (S)......Page 51
2.3.2 Singular White Noise......Page 53
2.4 The Wick Product......Page 54
2.4.1 Some Examples and Counterexamples......Page 58
2.5 Wick Multiplication and Hitsuda/Skorohod Integration......Page 61
2.6 The Hermite Transform......Page 72
2.7 The (S)N,r Spaces and the S-Transform......Page 86
2.8 The Topology of (S)-1N......Page 92
2.9 The F-Transform and the Wick Product on L1()......Page 99
2.10 The Wick Product and Translation......Page 103
2.11 Positivity......Page 109
3.1.1 Linear 1-Dimensional Equations......Page 125
3.1.2 Some Remarks on Numerical Simulations......Page 128
3.1.3 Some Linear Multidimensional Equations......Page 129
3.2 A Model for Population Growth in a Crowded,Stochastic Environment......Page 130
3.2.1 The General (S)-1 Solution......Page 131
3.2.2 A Solution in L1()......Page 133
3.2.3 A Comparison of Model A and Model B......Page 137
3.3 A General Existence and Uniqueness Theorem......Page 138
3.4 The Stochastic Volterra Equation......Page 141
3.5 Wick Products Versus Ordinary Products:a Comparison Experiment......Page 150
3.5.1 Variance Properties......Page 153
3.6 Solution and Wick Approximationof Quasilinear SDE......Page 155
3.7 Using White Noise Analysis to Solve GeneralNonlinear SDEs......Page 160
4.1 General Remarks......Page 168
4.2 The Stochastic Poisson Equation......Page 170
4.2.1 The Functional Process Approach......Page 172
4.3.1 Pollution in a Turbulent Medium......Page 173
4.4 The Stochastic Schrödinger Equation......Page 178
4.4.1 L1()-Properties of the Solution......Page 181
4.5 The Viscous Burgers Equation with a Stochastic Source......Page 187
4.6 The Stochastic Pressure Equation......Page 195
4.6.1 The Smoothed Positive Noise Case......Page 196
4.6.2 An Inductive Approximation Procedure......Page 201
4.6.3 The 1-Dimensional Case......Page 202
4.6.4 The Singular Positive Noise Case......Page 203
4.7 The Heat Equation in a Stochastic, Anisotropic Medium......Page 204
4.8 A Class of Quasilinear Parabolic SPDEs......Page 209
4.9 SPDEs Driven by Poissonian Noise......Page 212
5.1 Introduction......Page 222
5.2 The White Noise Probability Space of a Lévy Process (d=1)......Page 224
5.3.1 Chaos Expansion Theorems......Page 228
5.3.2 The Lévy--Hida--Kondratiev Spaces......Page 234
5.4.1 Construction of the Lévy Field......Page 241
5.4.2 Chaos Expansions and Skorohod Integrals (d1)......Page 247
5.4.3 The Wick Product......Page 253
5.4.4 The Hermite Transform......Page 255
5.5 The Stochastic Poisson Equation......Page 257
5.6 Waves in a Region with a Lévy White Noise Force......Page 261
5.7 Heat Propagation in a Domain with a LévyWhite Noise Potential......Page 262
Appendix A......Page 266
Appendix B......Page 271
Appendix C......Page 279
Appendix D......Page 281
Appendix E......Page 289
References......Page 297
List of frequently used notation and symbols......Page 304
Index......Page 309