Champagne subregions of the unit disc

TL;DR

This paper studies harmonic measure in the unit disc with infinitely many removed disjoint discs, focusing on configurations that Brownian motion cannot avoid, and refines existing results in this area.

Contribution

It provides new insights into which disc arrangements make certain boundary points unavoidable for Brownian motion, refining previous theoretical results.

Findings

Identifies configurations of removed discs that are unavoidable for Brownian motion

Refines existing theorems on harmonic measure and boundary behavior

Provides criteria for disc arrangements affecting harmonic measure

Abstract

This paper concerns harmonic measure on the domains that arise when infinitely many disjoint closed discs are removed from the unit disc. It investigates which configurations of discs are unavoidable for Brownian motion, and obtains refinements of related results of Akeroyd, and of Ortega-Cerd\`{a} and Seip.