PUB - Publikationen an der Universität Bielefeld

A champagne subdomain of a connected open set U not equal empty set in R-d, d >= 2, is obtained by omitting pairwise disjoint closed balls (B) over bar (x, r(x)), x is an element of 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 \ A and killed when leaving U, may hit A almost surely or, equivalently, A may have harmonic measure 1 for U \ A. Recent publications by Gardiner and Ghergu (d >= 3) and by Pres (d = 2) give rather sharp answers to the question of 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, which hold also for arbitrary connected open sets U. (C) 2013 Elsevier Inc. All rights reserved.