On the duration of stays of Brownian motion in domains in Euclidean space

Abstract

Let $T_D$ denote the first exit time of a Brownian motion from a domain $D$ in ${\mathbb{R}}^n$. Given domains $U,W\subseteq {\mathbb{R}}^n$ containing the origin, we investigate the cases in which we are more likely to have fast exits from $U$ than $W$, meaning $\mathbf{P}(T_U<t)>\mathbf{P}(T_W<t)$ for $t$ small. We show that the primary factor in the probability of fast exits from domains is the proximity of the closest regular part of the boundary to the origin. We also prove a result on the complementary question of longs stays, meaning $\mathbf{P}(T_U>t)>\mathbf{P}(T_W>t)$ for $t$ large. This result, which applies only in two dimensions, shows that the unit disk has the lowest probability of long stays amongst all Schlicht domains.