Polarization, continuous Markov processes, and second order elliptic equations

Abstract

We consider a continuous strong Markov process on a state space with a symmetry structure, and describe the behavior of its exit distribution, sojourn time, and transition function in an open set D when polarization is applied to D. This result generalizes known theorems for Brownian motion on Euclidean space. We apply the result to solutions of stochastic differential equations to obtain a comparison theorem for second-order elliptic partial differential equations.