ABSTRACT
This chapter analyses some parabolic equations in R domain. The analysis is based on the method of Fourier transform. The chapter presents some basic facts about the Fourier transforms of random fields and treats the existence and uniqueness questions for linear and semilinear parabolic equations. It gives a stochastic Feynman-Kac formula as a probabilistic representation of the solution for a linear parabolic equation with a multiplicative noise. The chapter studies the issue that solutions to a parabolic quation representing, say, the mass density, are required to be nonnegative on physical grounds. It provides the derivation of partial differential equations for the correlation functions of the solution to a certain linear parabolic equation.
