Weak Harnack Inequality and H\"older Regularity for Symmetric Stable L\'evy Processes

TL;DR

This paper establishes weak Harnack inequalities and H"older regularity for symmetric alpha-stable Levy processes in multi-dimensional space, under specific spectral measure conditions, advancing understanding of their regularity properties.

Contribution

It proves weak Harnack inequalities and H"older regularity for symmetric alpha-stable Levy processes with spectral measure assumptions, extending previous regularity results.

Findings

Weak Harnack inequality established for the processes.

H"older regularity results derived from the inequality.

Conditions on the spectral measure are crucial for the results.

Abstract

In this paper we consider weak Harnack inequality and H\"older regularity estimates for symmetric $\alpha$-stable L\'evy process in $\mathbb{R}^d$, $\alpha \in (0,2)$, $d\geq 2$. We consider a symmetric $\alpha$-stable L\'evy process $X$ 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, which we apply to prove H\"older regularity results.