A probabilistic proof of the boundary Harnack principle

Abstract

The main purpose of this paper is to give a probabilistic proof of Theorem 1.1, one using elementary properties of Brownian motion. We also obtain the fact that the Martin boundary equals the Euclidean boundary as an easy corollary of Theorem 1.1. The boundary Harnack principle may be viewed as a Harnack inequality for conditioned Brownian motion; as an application we prove some new probability bounds for conditioned Brownian motion in Lipschitz domains.