Random Walks Associated with Non-Divergence Form Elliptic Equations

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 C1, 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.