Asymptotic behaviour of a Brownian motion on exterior domains

Abstract

We study the asymptotic behaviour of the transition density of a Brownian motion in D, killed at @D, where Dc is a compact non polar set. Our main result concern dimension d D 2, where we show that the transition density pD t .x; y/ behaves, for large t , as 2 u.x/u.y/.t .log t/2/ −1 for x; y 2 D, where u is the unique positive harmonic function vanishing on .@D/r , such that u.x/ log jxj.