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arXiv2019

Weak Harnack Inequality and Hölder Regularity for Symmetric Stable Lévy Processes

In this paper we consider weak Harnack inequality and Hölder regularity estimates for symmetric $α$-stable Lévy process in $\mathbb{R}^d$, $α\in (0,2)$, $d\geq 2$. We consider a symmetric $α$-stable Lévy process $X$ for which a spherical part $μ$ of the Lévy measure is a spectral measure. In addition, we assume that $μ$ is absolutely continuous with respect to the uniform measure $σ$ on the sphere and impose certain bounds on the corresponding density. Eventually, we show that the weak Harnack inequality holds, which we apply to prove Hölder regularity results.

math.PR↗

arXiv2015

Harnack Inequalities for Symmetric Stable Levy Processes

In this paper we consider Harnack inequalities with respect to a symmetric $α$-stable Lévy process $X$ in $\mathbb{R}^d$, $α\in (0,2)$, $d\geq 2$. 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 Lévy measure. By checking the condition, in the case $α\in (0,1)$, 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 $α$-stable Lévy processes. We consider a symmetric $α$-stable Lévy process, $α\in (0,2)$, for which a spherical part $μ$ of the Lévy measure is a spectral measure. In addition, we assume that $μ$ is absolutely continuous with respect to the uniform measure $σ$ on the sphere and impose certain bounds on the corresponding density. Eventually, we show that the weak Harnack inequality holds.

math.PR↗