On Tsirelson's theorem about triple points for harmonic measure

TL;DR

This paper provides a new purely analytical proof of Tsirelson's 1997 theorem, which states that the set of triple points with mutually absolutely continuous harmonic measures in three disjoint domains has null harmonic measure.

Contribution

The paper introduces an analytical proof of Tsirelson's theorem, replacing the original probabilistic approach based on Brownian motion analysis.

Findings

The set of triple points with mutually absolutely continuous harmonic measures has null harmonic measure.

The proof is purely analytical, avoiding probabilistic methods.

The result applies to three disjoint domains in Euclidean space.

Abstract

A theorem of Tsirelson from 1997 asserts that given three disjoint domains in $\mathbb R^{n+1}$, the set of triple points belonging to the intersection of the three boundaries where the three corresponding harmonic measures are mutually absolutely continuous has null harmonic measure. The original proof by Tsirelson is based on the fine analysis of filtrations for Brownian and Walsh-Brownian motions and can not be translated into potential theory arguments. In the present paper we give a purely analytical proof of the same result.