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

TL;DR

This paper investigates how the shape and boundary proximity of domains in Euclidean space influence the likelihood of Brownian motion exiting quickly or staying long, revealing key geometric factors affecting these probabilities.

Contribution

It establishes the role of boundary proximity in exit times and characterizes the unit disk as minimizing long stay probabilities among Schlicht domains in two dimensions.

Findings

Proximity of boundary's regular part affects exit probabilities.

In two dimensions, the unit disk minimizes long stay probabilities among Schlicht domains.

Boundary geometry critically influences Brownian motion duration in domains.

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 ${\bf P}(T_U<t) > {\bf P}(T_W<t)$ 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 ${\bf P}(T_U>t) > {\bf P}(T_W>t)$ 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.