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

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).