arXiv 1503.051192015-03-17math.PR

对称稳定Levy过程的Harnack不等式

Harnack Inequalities for Symmetric Stable Levy Processes

Marina Sertic

摘要：

本文考虑关于对称$α$ -稳定Lévy过程$X$在$\mathbb{R}^d$，$α\in (0,2)$，$d\geq 2$中的Harnack不等式.我们研究文章\cite{bg-sz-1}中的示例。在那里，作者将Harnack不等式与相对Kato条件联系起来，这是Lévy测度的一个条件。通过检查条件，在$α\in (0,1)$的情况下，他们已经建立了Harnack不等式不成立。我们给这个事实的另一种证明，使用设置的\cite{bg-sz-1}。我们明确地定义了调和函数。对于一个给定的起始点的过程中，我们研究的概率击中一定的设置在第一次退出时间的一个单位球。此外，我们还研究了一类对称$α$ -稳定Lévy过程的弱Harnack不等式.考虑对称$α$稳定Lévy过程$α\in (0,2)$，其中Lévy测度的球面部分$μ$是谱测度.另外，我们假设$μ$关于球面上的一致测度$σ$是绝对连续的，并对相应的密度施加一定的界限.最后证明了弱Harnack不等式成立.

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文