The transition density of Brownian motion killed on a bounded set

We study the transition density of a standard two-dimensional Brownian motion killed when hitting a bounded Borel set $A$. We derive the asymptotic form of the density, say $p^A_t({\bf x},{\bf y})$, for large times $t$ and for ${\bf x}$ and ${\bf y}$ in the exterior of $A$ valid uniformly under the constraint $|{\bf x}|\vee |{\bf y}| =O(t)$. Within the parabolic regime $|{\bf x}|\vee |{\bf y}| = O(\sqrt t)$ in particular $p^A_t({\bf x},{\bf y})$ is shown to behave like $4e_A({\bf x})e_A({\bf y}) (\lg t)^{-2} p_t({\bf y}-{\bf x})$ for large $t$, where $p_t({\bf y}-{\bf x})$ is the transition kernel of the Brownian motion (without killing) and $e_A$ is the Green function for the \lq exterior of $A$' with a pole at infinity normalized so that $e_A({\bf x}) \sim \lg |{\bf x}|$. We also provide fairly accurate upper and lower bounds of $p^A_t({\bf x},{\bf y})$ for the case $|{\bf x}|\vee |{\bf y}|>t$ as well as corresponding results for the higher dimensions.