Asymptotic behaviour of a Brownian motion on exterior domains

Abstract

We study the asymptotic behaviour of the transition density of a Brownian motion in script D, killed at ∂script D, where script Dc is a compact non polar set. Our main result concern dimension d = 2, where we show that the transition density pscript Dt (x, y) behaves, for large t, as 2/πu(x)u(y)(t(logt)2)-1 for x, y ∈ script D, where u is the unique positive harmonic function vanishing on (∂script D)r, such that u(x) ∼ log |x|.