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 $p_t^A(\mathbf{x},\mathbf{y})$, for large times $t$ and for ${\bf x}$and${\bf y}$intheexteriorof$A$validuniformlyundertheconstraint$|{\bf x}|\vee |{\bf y}| =O(t)$\mathrm{.}Withintheparabolicregime$|{\bf x}|\vee |{\bf y}| = O(\sqrt t)$inparticular$p^A_t({\bf x},{\bf y})$ is shown to behave like $4e_A(\mathbf{x})e_A(\mathbf{y})(\lg t{)}^{-2}{p}_t(\mathbf{y}-\mathbf{x})$ for large $t$, where $p_t(\mathbf{y}-\mathbf{x})$ 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 $e_A({\bf x}) \sim \lg |{\bf x}|$\mathrm{.}Wealsoprovidefairlyaccurateupperandlowerboundsof$p^A_t({\bf x},{\bf y})$forthecase$|{\bf x}|\vee |{\bf y}|>t$ as well as corresponding results for the higher dimensions.