Linear Random Operator Equations in Mathematical Physics. II

Abstract

This paper is a continuation of Paper I [J. Math. Phys. 11, 1069 (1970)]. A general expression is determined for the stochastic Green's function (SGF) for two-point correlation functions, and various useful relationships are determined between the stochastic Green's functions for various statistical measures and between the stochastic Green's functions, random Green's functions, and ordinary Green's functions. In the author's dissertation and earlier papers, SGF was given as an ensemble average of the product of random Green's functions. This random Green's function is now specified in terms of an ordinary Green's function for a deterministic operator and a resolvent kernel which can be calculated for the random part of the stochastic operator. Hence that SGF is determinable which yields the desired statistical measure of the solution process directly. Second, the two-point correlation function of the solution process is found for the perturbation case. It is also demonstrated that, in the event that perturbation theory is adequate to deal with the randomness involved, the correct two-point correlation of the solution process is easily specialized from the general expression, i.e., the results of perturbation theory are obtained from the SGF of Adomian when perturbation theory is applicable.