Harnack Inequalities for Symmetric Stable Levy Processes

TL;DR

This paper investigates Harnack inequalities for symmetric alpha-stable Levy processes, providing new proofs and conditions under which these inequalities hold or fail, especially focusing on the case alpha in (0,1).

Contribution

The paper offers an alternative proof that Harnack inequalities do not hold for alpha in (0,1) and establishes conditions under which the weak Harnack inequality is valid for symmetric alpha-stable Levy processes.

Findings

Harnack inequality does not hold for alpha in (0,1).

Weak Harnack inequality holds under certain spectral measure conditions.

Explicit harmonic functions are constructed for the processes.

Abstract

In this paper we consider Harnack inequalities with respect to a symmetric $\alpha$-stable L\'evy process $X$ in $\mathbb{R}^d$, $\alpha \in (0,2)$, $d\geq 2$. We study the example from the article \cite{bg-sz-1}. There, the authors have associated the Harnack inequality with the relative Kato condition, which is a condition on the L\'evy measure. By checking the condition, in the case $\alpha \in (0,1)$, they have established that the Harnack inequality does not hold. We give an alternative proof of this fact, using the setting of \cite{bg-sz-1}. We define the harmonic functions explicitly. For a given starting point of the process, we examine the probability of hitting a certain set at the first exit time of a unit ball. Moreover, we also examine the weak Harnack inequality for a certain class of symmetric $\alpha$-stable L\'evy processes. We consider a symmetric $\alpha$-stable L\'evy process, $\alpha \in (0,2)$, for which a spherical part $\mu$ of the L\'evy measure is a spectral measure. In addition, we assume that $\mu$ is absolutely continuous with respect to the uniform measure $\sigma$ on the sphere and impose certain bounds on the corresponding density. Eventually, we show that the weak Harnack inequality holds.