On Brownian Excursions in Lipschitz Domains. Part II: Local Asymptotic Distributions

Abstract

In this paper, we continue the study initiated in Burdzy and Williams (1986) of the local properties of Brownian excursions in Lipschitz domains. The focus in part I was on local path properties of such excursions. In particular, a necessary and sufficient condition was given for Brownian excursions in a Lipschitz domain to share the local path properties with Brownian excursions in a half-space. This condition holds for C [to the power of 1, alpha] -domains ([alpha] > 0), but there is a C [to the power of 1]-domain for which it fails. Here we consider the distributions of a selection of local events for excursions. In particular, we focus on the asymptotics of these distributions as the region of locality shrinks to a point. We show that when a Lipschitz domain is locally approximated by a half-space, the asymptotics for excursions in the two domains are comparable.