Brownian motion in a ball in the presence of spherical obstacles

TL;DR

This paper establishes a precise integral condition determining when Brownian motion within a unit ball can avoid a countable set of spherical obstacles, advancing understanding of stochastic processes in constrained environments.

Contribution

It provides a necessary and sufficient integral criterion for the avoidability of spherical obstacles by Brownian motion in a bounded domain.

Findings

Derived a clear integral condition for obstacle avoidability.

Characterized the probabilistic behavior of Brownian motion amidst obstacles.

Enhanced theoretical understanding of stochastic avoidance in geometric settings.

Abstract

We study the problem of when a Brownian motion in the unit ball has a positive probability of avoiding a countable collection of spherical obstacles. We give a necessary and sufficient integral condition for such a collection to be avoidable.