Champagne subdomains with unavoidable bubbles
Wolfhard Hansen, Ivan Netuka

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 in , , is obtained omitting pairwise disjoint closed balls , , the bubbles, where is an infinite, locally finite set in . The union of these balls may be unavoidable, that is, Brownian motion, starting in and killed when leaving , may hit almost surely or, equivalently, may have harmonic measure one for . Recent publications by Gardiner/Ghergu () and by Pres () give rather sharp answers to the question how small such a set may be, when 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 .
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Taxonomy
TopicsMathematical Dynamics and Fractals · Analytic and geometric function theory · Geometric Analysis and Curvature Flows
