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

Abstract

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 for X and a decreasing function g : [0, ∞) → (0, ∞] having the doubling property such that G ≈ g•ρ.Assuming that the constant function 1 is harmonic and balls are relatively compact, is is shown that every positive harmonic function 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 z∈Z g(ρ(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. Vondraček, and the author. . MSC: 31B15, 31C15, 31D05, 60J25, 60J45, 60J65, 60J75.ρ(x, y) = |x − y|, and g(r) = r α−d , then (4.6) means thatwhich, in the classical case α = 2, is the separation property in [3, Theorem 6].THEOREM 4.6. Let A be an avoidable union of pairwise disjoint balls B(z, r z ), z ∈ Z ⊂ X, having the separation property with respect to λ. ThenProof. We may suppose that ρ(x 0 , z) > 4R 0 , for every z ∈ Z (we simply omit finitely many points from Z). Moreover, we may assume without loss of generality that (4.7) r z ≤ ρ(x 0 , z)/2, for every z ∈ Z.Indeed, replacing r z by r ′ z := min{r z , ρ(x 0 , z)/2} our assumptions are preserved. Suppose we have shown that z∈Z g(ρ(x 0 , z))/g(r ′ z ) < ∞. Since g(r)/g(r/2) ≥ c −1

On Evans’ and Choquet’s Theorems for Polar Sets

“…In particular, the local triangle property holds for the classical Green function not only for domains in R N , N ≥ 3, but also on domains X in R 2 such that R 2 \ X is not polar (log(2/r) ≤ 2 log(1/r) for 0 < r ≤ 1/2) and as well for Green functions associated with very general Lévy processes (see [12]).…”

Scaling invariant Harnack inequalities in a general setting

“…SinceG = ∞ on the diagonal, (7.2) implies thatG(y, x) ≤cG(x, y) and (x, y) →G(x, y) −1 +G(y, x) −1 defines a quasi-metric on X which is equivalent toG −1 . So, by [15,Proposition 14.5] (see also [11,Proposition 6.1]), there exist a metricρ on X, γ ≥ 1 and C ≥ 1 such that (7.5) holds. Now let x ∈ X, r > 0 and s := C −1 r γ .…”