Transition densities of reflecting Brownian motions on Lipschitz domains

TL;DR

This paper investigates the transition density's continuity and local estimates for reflecting Brownian motion on Lipschitz domains, establishing the surface measure's inclusion in the local Kato class.

Contribution

It provides new continuity results and local density estimates for reflecting Brownian motion on Lipschitz domains, linking surface measure to the Kato class.

Findings

Transition density is continuous on Lipschitz domains.

Local estimates for the transition density are established.

Surface measure belongs to the local Kato class for reflecting Brownian motion.

Abstract

In this paper, we study the continuity of the transition density of the reecting Brownian motion on a general Lipschitz domain. We also provide local estimates for the density. Applying the estimates, we prove that the surface measure on the domain is in the local Kato class of the reecting Brownian motion.