Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach (Hardcover, 1996)

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-Ito chaos expansion.- Chaos expansion in terms of Hermite polynomials.- Chaos expansion in terms of multiple Ito 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 Ito/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 Schrodinger 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 Ito calculus.- The Ito 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.