Harnack Inequalities for Symmetric Stable Levy Processes

الأصل · EN

In this paper we consider Harnack inequalities with respect to a symmetric α-stable Lévy process X in Rᵈ, α∈ (0,2), d≥ 2. We study the example from the article 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∈ (0,1), they have established that the Harnack inequality does not hold. We give an alternative proof of this fact, using the setting of 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∈ (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.