The Transition Density of Brownian Motion Killed on a Bounded Set

Abstract

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 $p^A_t(\mathbf{x},\mathbf{y})$, for large times t and for $\mathbf{x}$ and $\mathbf{y}$ in the exterior of A valid uniformly under the constraint $|\mathbf{x}|\vee |\mathbf{y}| =O(t)$. Within the parabolic regime $|\mathbf{x}|\vee |\mathbf{y}| = O(\sqrt{t})$ in particular $p^A_t(\mathbf{x},\mathbf{y})$ 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 ‘exterior of A’ with a pole at infinity normalized so that $e_A(\mathbf{x}) \sim \lg |\mathbf{x}|$. We also provide fairly accurate upper and lower bounds of $p^A_t(\mathbf{x},\mathbf{y})$ for the case $|\mathbf{x}|\vee |\mathbf{y}|>t$ as well as corresponding results for the higher dimensions.