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≥q 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≥q 3 on which there exists a nonconstant Dirichlet finite harmonic measure.