Trapping planar Brownian motion in a non circular trap

TL;DR

This paper investigates how planar Brownian motion interacts with a non-circular trap, demonstrating that hitting probabilities can be effectively approximated by a disk with radius equal to the trap's conformal radius.

Contribution

It introduces a method to approximate hitting probabilities for arbitrary connected traps using conformal radius, extending understanding beyond circular traps.

Findings

Hitting probability approximations are accurate even at short times.

Conformal radius effectively characterizes the trap's influence on Brownian motion.

The approach generalizes previous results limited to circular traps.

Abstract

Brownian motion in the plane in the presence of a "trap" at which motion is stopped is studied. If the trap $T$ is a connected compact set, it is shown that the probability for planar Brownian motion to hit this set before a given time $t$ is well approximated even at short times by the probability that Brownian motion hits a disk of radius $r_T$ equal to the conformal radius of the trap $T$.