Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach(Probability and its Applications)

The main emphasis of this work is on stochastic partial differential equations. First the stochastic Poisson equation and the stochastic transport equation are discussed; then the authors go on to deal with the Schrodinger equation, the heat equation, the nonlinear Burgers' equation with a stochastic source, and the pressure equation. The white noise approach often allows for solutions given by explicit formulas in terms of expectations of certain auxiliary processes. The noise in the above examples are all of a Gaussian white noise type. In the end, the authors also show how to adapt the analysis to SPDEs involving noise of Poissonian type.

Table of Contents:
Introduction - modelling by stochastic differential equations. Framework - white noise: the 1-dimensional, d-parameter (smoothed) white noise, the (smoothed) white noise vector; the Wiener-Ito chaos expansion: expansion in terms of hermite polynomials, chaos expansion in terms of multiple Ito integrals; the Wick product: some examples and counterexamples; Wick multiplication and Ito/Skorohod integration; the Hermite transform: definition, relation to wick product; the Wick product and translation; positivity. Applications to stochastic ordinary differential equations - linear equations: linear 1-dimensional equations, some linear multi-dimensional equations; a model for population growth in a crowded, stochastic environment; a general existence and uniqueness theorem: the general linear multi-dimensional equation; the stochastic Volterra equation; Wick product versus ordinary product: a comparison experiment; solution and Wick approximation of quasilinear SDE. Stochastic partial differential equations - general remarks; the stochastic Poisson equation: the functional process approach; the stochastic transport equation: pollution in a turbulent medium, the heat equation with a stochastic potential; the viscous Burgers equation with a stochastic source; the stochastic pressure equation: the smoothed positive noise case, an inductive approximation procedure, the 1-dimension case, the singular positive noise case; the heat equation in a stochastic anisotropic medium; a class of quasilinear parabolic SPDEs; SPDEs driven by Poissonian noise. (Part contents).Poissonian noise. (Part contents).