Toward a stochastic calculus for several Markov processes

Abstract

In this paper we investigate a class of harmonic functions associated with a pair x t = ( x t 1 1, x t 2 2) of strong Markov processes. In the case where both processes are Brownian motions, a smooth function f is harmonic if Δ x 1 Δ x 2 f( x 1, x 2) = 0. For these harmonic functions we investigate a certain boundary value problem which is analogous to the Dirichlet problem associated with a single process. One basic tool for this study is a generalization of Dynkin's formula, which can be thought of as a kind of stochastic Green's formula. Another important tool is the use of Markov processes x t i − i obtained from x t i i by certain random time changes. We call such a process a stochastic wave since it propogates deterministically through a certain family of sets; however its position on a given set is random.