Green's Functions for Elliptic and Parabolic Equations with Random Coefficients Ii

This paper is concerned with linear parabolic partial diierential equations in divergence form and their discrete analogues. It is assumed that the coeecients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously diierentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related nite diierence equations. This paper generalises results of a previous paper which considered the case when the coeecients are constant in time but random in space.