The transition density of Brownian motion killed on a bounded set

TL;DR

This paper derives the asymptotic behavior of the transition density of a two-dimensional Brownian motion killed upon hitting a bounded set, providing detailed estimates for large times and spatial regimes.

Contribution

It introduces new asymptotic formulas for the transition density of killed Brownian motion in two dimensions, including uniform bounds and behavior in different spatial regimes.

Findings

Asymptotic form of the transition density for large times

Behavior of the density in the parabolic regime

Upper and lower bounds for the density outside the parabolic regime

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 $p^A_t({\bf x},{\bf y})$, 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.