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Unavoidable sets and Wiener's test for Hunt processes

Let (X,W) be a balayage space, 1 ∈ W, or – equivalently – let W be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that W separates points, every function in W is the supremum of its continuous minorants and there exist strictly positive continuous u, v ∈ W such that u/v → 0 at infinity. We suppose that there is a Green function G > 0 for X, a metric ρ on X and a decreasing function g : [0,∞) → (0,∞] having the doubling property and a mild upper decay at infinity such that G ≈ g ◦ ρ (which is equivalent to a 3G-inequality). Then the corresponding capacity for balls of radius R is bounded by a constant multiple of 1/g(R). Assuming that the constant function 1 is harmonic and the capacity of large balls satisfies a reverse estimate or that bounded functions are harmonic if and only if they are constant (Liouville property), it is proven 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, rz), 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 x0 ∈ X, the series ∑ z∈Z g(ρ(x0, z))/g(rz) 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. Vondraček, and the author.