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 $p_t^A(\mathbf{x},\mathbf{y})$, 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 $p_t^A(\mathbf{x},\mathbf{y})$ 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 \lq 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 $p_t^A(\mathbf{x},\mathbf{y})$ for the case $|\mathbf{x}|\vee |\mathbf{y}|>t$ as well as corresponding results for the higher dimensions.

arXiv comment25 pages, to appear in Journal of Theoretical Probability