Champagne subdomains with unavoidable bubbles

TL;DR

This paper investigates champagne subdomains created by removing infinitely many disjoint balls from a connected open set, focusing on when these sets are unavoidable for Brownian motion, and provides optimal results applicable to any such domain.

Contribution

It introduces a novel approach to determine the minimal size of unavoidable sets in champagne subdomains, extending results beyond the unit ball to arbitrary connected open sets.

Findings

Established optimal size bounds for unavoidable sets in champagne subdomains.

Extended results to arbitrary connected open sets, not just the unit ball.

Provided a new method differing from previous approaches.

Abstract

A champagne subdomain 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 an infinite, 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 totally different approach, optimal results are obtained, results which hold as well for arbitrary connected open sets $U$.