Asymptotic behaviour of a Brownian motion on exterior domains

We study the asymptotic behaviour of the transition density of a Brownian m otion in D, killed at partial derivative D, where D-c is a compact non pola r set. Our main result concern dimension d = 2, where we show that the tran sition density p(t)(D)(x, y) behaves, for large t, as 2/pi u(x)u(y)(t(log t )(2))(-1) for x, y is an element of D, where u is the unique positive harmo nic function vanishing on (partial derivative D)(r), such that u(x) similar to log \x\.