Champagne subregions with unavoidable bubbles

TL;DR

This paper investigates the conditions under which a union of disjoint balls in a domain can be unavoidable for Brownian motion, providing stronger, more general results than previous studies.

Contribution

It introduces a new criterion for unavoidable sets and extends sharp results to arbitrary open sets, not just the unit ball.

Findings

Stronger conditions for unavoidable sets are established.

Results apply to any open set, not limited to the unit ball.

New criterion simplifies the analysis of unavoidable sets.

Abstract

A champagne subregion of a connected open set $U\ne\emptyset$ in $R^d$, $d\ge 2$, is obtained omitting pairwise disjoint closed balls $\bar B(x, r_x)$, $x\in X$, the bubbles, where $X$ is a locally finite set in $U$. The union $A$ of these balls may be unavoidable, that is, Brownian motion, starting in $U\setminus A$ and killed when leaving $U$, may hit $A$ almost surely or, equivalently, $A$ may have harmonic measure one for $U\setminus A$. Recent publications by Gardiner/Ghergu ($d\ge 3$) and by Pres ($d=2$) give rather sharp answers to the question how small such a set $A$ may be, when $U$ is the unit ball. In this paper, using a new criterion for unavoidable sets and a straightforward approach, much stronger results are obtained, results which hold as well for an arbitrary open set $U$.