Stochastic Partial Differential Equations

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 norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy". We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corre­ sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in reservoir theory and related areas. 3) The theory should be strong and efficient enough to allow us to solve th,~se SPDEs explicitly, or at least provide algorithms or approximations for the solutions.

Contenu

1. Introduction.- 1.1. Modeling by stochastic differential equations.- 2. Framework.- 2.1. White noise.- The 1-dimensional, d-parameter smoothed white noise.- The (smoothed) white noise vector.- 2.2. The Wiener-Itô chaos expansion.- Chaos expansion in terms of Hermite polynomials.- Chaos expansion in terms of multiple Itô integrals.- 2.3. Stochastic test functions and stochastic distributions.- The Kondratiev spaces (S)pm;N, (S)-pm;N.- The Hida test function space(S) and the Hida distribution space(S)*.- Singular white noise.- 2.4. The Wick product.- Son e examples and counterexamples.- 2.5. Wick multiplication and Itô/Skorohod integration.- 2.6. The Hermite transform.- Tht characterization theorem for(S)-1N.- Positive noise.- The positive noise matrix.- 2.7. The S)p,rN spaces and the S-transform.- The S-transform.- 2.8. The topology of (S)-1N.- Stochastic distribution processes.- 2.9. The F-transform and the Wick product on L1 (?).- Functional processes.- 2.10. The Wick product and translation.- 2.11. Positivity.- Exercises.- 3. Applications to stochastic ordinary differential equations.- 3.1. Linear equations.- Linear 1-dimensional equations.- Some remarks on numerical simulations.- Some linear multi-dimensional equations.- 3.2. A model for population growth in a crowded stochastic environment.- The general(S)-1 solution.- A solution in L1(?).- A comparison of Model A and Model B.- 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.- Exercises.- 4. Stochastic partial differential equations.- 4.1. General remarks.- 4.2. The stochastic Poisson equation.- The functional process approach.- 4.3. The stochastic transport equation.- Pollution in a turbulent medium.- The heat equation with a stochastic potential.- 4.4. The stochastic Schrödinger equation.- L1 (?)-properties of the solution.- 4.5. The viscous Burgers' equation with a stochastic source.- 4.6. The stochastic pressure equation.- The smoothed positive noise case.- An inductive approximation procedure.- The 1-dimensional case.- The singular positive case.- 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.- Exercises.- 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.