Random Walks Associated with Non-Divergence Form Elliptic Equations

Abstract

This paper is concerned with the study of the diffusion process associated with a nondivergence form elliptic operator in d dimensions, d ≥2. The authors introduce a new technique for studying the diffusion, based on the observation that the probability of escape from a d −1 dimensional hyperplane can be explicitly calculated. They use the method to estimate the probability of escape from d −1 dimensional manifolds which are C 1, α , and also d −1 dimensional Lipschitz manifolds. To implement their method the authors study various random walks induced by the diffusion process, and compare them to the corresponding walks induced by Brownian motion.