Dirichlet finite harmonic measures on topological balls

Abstract

Based upon an intuition from electrostatics one might suspect that there is no topological ball in Euclidean space of dimension $d\ge 2$ which carries a nonconstant Dirichlet finite harmonic measure. This guess is certainly true for $d=2$. However, contrary to the above intuition, it is shown in this paper that there does exist a topological ball in Euclidean space of every dimension $d\ge 3$ on which there exists a nonconstant Dirichlet finite harmonic measure.