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

摘要

This paper is concerned with linear uniformly elliptic and parabolic partialdifferential equations in divergence form. It is assumed that the coefficientsof the equations are random variables, constant in time. The Green's functionsfor the equations are then random variables. Regularity properties forexpectation values of Green's functions are obtained. In particular, it isshown that the expectation value is a continuously differentiable functionwhose derivatives are bounded by the corresponding derivatives of the heatequation. Similar results are obtained for the related finite differenceequations.