A Martin Boundary in the Plane

Abstract

Let $E$ be an open connected subset of Euclidean space, with a Green function, and let $\lambda$ be harmonic measure on the Martin boundary $\Delta$ of $E$. We will show that, except for a $\lambda \otimes \lambda$-null set of $(x,y) \in {\Delta ^2}$, $x$ is an entrance point for Brownian motion conditioned to leave $E$ at $y$. R. S. Martin gave examples in dimension $3$ or higher, for which there exist minimal accessible Martin boundary points $x \ne y$ for which this condition fails. We will give a similar example in dimension $2$.