Based upon an intuition from electrostatics one might suspect that there is no topological ball in Euclidean space of dimension which carries a nonconstant Dirichlet finite harmonic measure. This guess is certainly true for . 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 on which there exists a nonconstant Dirichlet finite harmonic measure.
Mitsuru NAKAI. "Dirichlet finite harmonic measures on topological balls." J. Math. Soc. Japan 52 (3) 501 - 513, July, 2000. https://doi.org/10.2969/jmsj/05230501