Book Description
At the same time there has been considerable development in the mathematical theory of stochastic differential equations which are used to model these phenomena. In this book the authors give a comprehensive introduction to stochastic partial differential equations. Their approach is based on white noise analysis, where the often ill-defined white noise, the derivative of the familiar Brownian motion, is introduced rigorously as the fundamental object.
First some of the mathematical background is discussed to provide the necessary tools to study several different stochastic partial differential equations. The techniques are primarily derived from functional analysis. The Wiener-Itô chaos expansion as well as the Itô/Skorohod integrals are developed in this setting, and properties of the Wick product and the Hermite transform are proved. The first applications are given to stochastic ordinary differential equations, e.g., the Volterra equation.
The main emphasis of the book is on stochastic partial differential equations. First the stochastic Poisson equation and the stochastic transport equation are discussed. Next, the authors consider the stochastic Schrödinger equation as well as the stochastic heat equation. The nonlinear Burgers' equation with a stochastic source is discussed, and finally the stochastic pressure equation, as well as other important equations are treated. 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 Gaussian white noise type, but in the end the authors also show how to adapt the analysis to SPDEs involving noise of Poissonian type.
