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

Abstract

Let \((X,\mathcal {W})\) be a balayage space, \(1\in \mathcal {W}\), or – equivalently – let \(\mathcal W\) be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that \(\mathcal {W}\) separates points, every function in \(\mathcal {W}\) is the supremum of its continuous minorants and there exist strictly positive continuous \(u,v\in \mathcal {W}\) such that u/v → 0 at infinity. We suppose that there is a Green function G > 0 for X, a metric ρ for X and a decreasing function \(g\colon [0,\infty )\to (0,\infty ]\) having the doubling property such that G ≈ g ∘ ρ. Assuming that the constant function 1 is harmonic and balls of (X, ρ) 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 ∈ Z, which have a certain separation property with respect to a suitable measure λ on X are unavoidable if and only if, for some/any point x 0 ∈ X, the series \({\sum }_{z\in Z} g(\rho (x_{0},z))/g(r_{z}) \) 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. Vondracek, and the author.