Harnack inequalities for Hunt processes with Green function

TL;DR

This paper establishes Harnack inequalities for positive harmonic functions associated with Hunt processes that have a Green function, under certain geometric and analytic conditions, extending results to a broad class of Levy processes.

Contribution

It introduces conditions under which Harnack inequalities hold for Hunt processes with Green functions, generalizing previous results to more complex Levy processes.

Findings

Harnack inequalities hold under specified Green function and capacity conditions.

Capacity bounds for balls are proportional to the inverse of a doubling function.

Sufficient conditions are provided for processes with jumps to satisfy Harnack inequalities.

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 and a mild upper decay near $0$ such that $G\approx g\circ\rho$ (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 reverse inequalities hold as well and that jumps of the process, when starting at neighboring points, are related in a suitable way, it is proven that positive harmonic functions satisfy scaling invariant Harnack inequalities. Provided that the Ikeda-Watanabe formula holds, sufficient conditions for this relation are given. This shows that rather general L\'evy processes are covered by this approach.