Champagne subregions with unavoidable bubbles

Abstract

A champagne subregion of a connected open set $U\ne \varnothing$ 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 $U$.

10 pages

References in corpus (1)

• Configurations of balls in Euclidean space that Brownian motion cannot avoid