Hunt's hypothesis (H) and the triangle property of the Green function

TL;DR

This paper proves that Hunt's hypothesis (H) holds for certain balayage spaces on locally compact abelian groups with a Green function satisfying a local triangle inequality, extending results for many Lévy processes.

Contribution

It establishes Hunt's hypothesis (H) under the condition of a Green function satisfying a local triangle inequality, linking potential theory and Lévy processes.

Findings

Hunt's hypothesis (H) holds under the given conditions.

The Green function satisfies a local triangle inequality for many Lévy processes.

The results connect the structure of balayage spaces with Hunt's hypothesis.

Abstract

Let $X$ be a locally compact abelian group with countable base and let $\mathcal W$ be a convex cone of positive numerical functions on $X$ which is invariant under the group action and such that $(X,\mathcal W)$ is a balayage space or (equivalently, if $1\in \mathcal W$) such that $\mathcal W$ is the set of excessive functions of a Hunt process on $X$, $\mathcal W$ separates points, every function in $\mathcal W$ is the supremum of its continuous minorants in $\mathcal W$, and there exist strictly positive continuous $u,v\in \mathcal W$ such that $u/v\to 0$ at infinity. Assuming that there is a Green function $G>0$ for $X$ which locally satisfies the triangle inequality $G(x,z)\wedge G(y,z)\le C G(x,y)$ (true for many L\'evy processes), it is shown that Hunt's hypothesis (H) holds, that is, every semipolar set is polar.