Unavoidable sets and harmonic measures living on small sets

Given a connected open set U not equal theta in R-d, d >= 2, a relatively closed set A in U is called unavoidable in U if Brownian motion, starting in x is an element of 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 proved, 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) over bar (z, r(z)), z is an element of Z, satisfying sup(z is an element of Z) r(z)/dist(z, U-c) < 1), a combination of the criterion and the existence of small non-polar 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 also for Riesz potentials (isotropic alpha-stable processes) on Euclidean space and for censored stable processes on C-1,C-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 Levy 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.