Liouville property, Wiener's test and unavoidable sets for Hunt processes

TL;DR

This paper establishes conditions under which positive harmonic functions are constant and characterizes unavoidable sets for Hunt processes using Green functions, Wiener's test, and geometric properties of the space.

Contribution

It generalizes previous results by providing new criteria for Liouville property and unavoidable sets in balayage spaces with Green functions, simplifying earlier complex proofs.

Findings

Liouville property holds under specified conditions.

Wiener's test characterizes unavoidable sets via series divergence.

Unions of disjoint balls are unavoidable if a series involving Green functions diverges.

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 $u/v\to 0$ at infinity. We suppose that there is a Green function $G>0$ for $X$, a metric $\rho$ on $X$ and a decreasing function $g\colon[0,\infty)\to (0,\infty]$ having the doubling property such that $G\approx g\circ\rho$. 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 $B(z,r_z)$, $z\in Z$, 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 $x_0\in X$, 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. Vondra\v cek, and the author.