Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes

摘要

Let be a balayage space, , or - equivalently - let be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that separates points, every function in is the supremum of its continuous minorants and there exist strictly positive continuous such that u/v -> 0 at infinity. We suppose that there is a Green function G > 0 for X, a metric rho for X and a decreasing function having the doubling property such that G approximate to g omicron rho. Assuming that the constant function 1 is harmonic and balls of (X, rho) are relatively compact, it is shown that every positive harmonic function on X is constant (Liouville property) and that Wiener's test at infinity shows, if a given set A in X is unavoidable, that is, if the process hits A with probability one, wherever it starts. An application yields that locally finite unions of pairwise disjoint balls B(z, r (z) ), z a Z, which have a certain separation property with respect to a suitable measure lambda on X are unavoidable if and only if, for some/any point x (0) a X, the series diverges. The results generalize and, exploiting a zero-one law for hitting probabilities, simplify recent work by S. Gardiner and M. Ghergu, A. Mimica and Z. Vondraek, and the author.