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Stochastic Partial Differential Equations A Modeling, White Noise Functional Approach

This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera­ tion between the Norwegian Academy of Science and Letters and Den n...

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Bibliographic Details
Main Authors: Holden, Helge, Oksendal, Bernt (Author), Uboe, Jan (Author), Zhang, Tusheng (Author)
Format: eBook
Language:English
Published: Boston, MA Birkhäuser 1996, 1996
Edition:1st ed. 1996
Series:Probability and Its Applications
Subjects:
Probability Theory
Mathematical Physics
Differential Equations
Theoretical, Mathematical And Computational Physics
Probabilities
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 1. Introduction
  • 1.1. Modeling by stochastic differential equations
  • 2. Framework
  • 2.1. White noise
  • 2.2. The Wiener-Itô chaos expansion
  • 2.3. Stochastic test functions and stochastic distributions
  • 2.4. The Wick product
  • 2.5. Wick multiplication and Itô/Skorohod integration
  • 2.6. The Hermite transform
  • 2.7. The S)p,rN spaces and the S-transform
  • 2.8. The topology of (S)-1N
  • 2.9. The F-transform and the Wick product on L1 (?)
  • 2.10. The Wick product and translation
  • 2.11. Positivity
  • 3. Applications to stochastic ordinary differential equations
  • 3.1. Linear equations
  • 3.2. A model for population growth in a crowded stochastic environment
  • 3.3. A general existence and uniqueness theorem
  • 3.4. The stochastic Volterra equation
  • 3.5. Wick products versus ordinary products: A comparison experiment Variance properties
  • 3.6. Solution and Wick approximation of quasilinear SDE
  • 4. Stochastic partial differential equations
  • 4.1. General remarks
  • 4.2. The stochastic Poisson equation
  • 4.3. The stochastic transport equation
  • 4.4. The stochastic Schrödinger equation
  • 4.5. The viscous Burgers’ equation with a stochastic source
  • 4.6. The stochastic pressure equation
  • 4.7. The heat equation in a stochastic, anisotropic medium
  • 4.8. A class of quasilinear parabolic SPDEs
  • 4.9. SPDEs driven by Poissonian noise
  • Appendix A. The Bochner-Minlos theorem
  • Appendix B. A brief review of Itô calculus
  • The Itô formula
  • Stochastic differential equations
  • The Girsanov theorem
  • Appendix C. Properties of Hermite polynomials
  • Appendix D. Independence of bases in Wick products
  • References
  • List of frequently used notation and symbols

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