Unavoidable sets and harmonic measures living on small sets

Given a connected open set U 6= ∅ in Rd, d ≥ 2, a relatively closed set A in U is called unavoidable in U , if Brownian motion, starting in x ∈ U \A and killed when leaving U , hits A almost surely or, equivalently, if the harmonic measure for x with respect to U \ A has mass 1 on A. First a new criterion for unavoidable sets is proven which facilitates the construction of smaller and smaller unavoidable sets in U . Starting with an arbitrary champagne subdomain of U (which is obtained omitting a locally finite union of pairwise disjoint closed balls B(z, rz), z ∈ Z, satisfying supz∈Z rz/dist(z, U c) < 1), a combination of the criterion and the existence of small nonpolar compact sets of Cantor type yields a set A on which harmonic measures for U \A are living and which has Hausdorff dimension d − 2 and, if d = 2, logarithmic Hausdorff dimension 1. This can be done as well for Riesz potentials (isotropic α-stable processes) on Euclidean space and for censored stable processes on C1,1 open subsets. Finally, in the very general setting of a balayage space (X,W) on which the function 1 is harmonic (which covers not only large classes of second order partial differential equations, but also non-local situations as, for example, given by Riesz potentials, isotropic unimodal Lévy processes or censored stable processes) a construction of champagne subsets X \ A of X with small unavoidable sets A is given which generalizes (and partially improves) recent constructions in the classical case.