Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes

Simple item page
dc.creatorChen, Zhen-Qing
dc.creatorSong, Renming
dc.date1998-09-29
dc.date.accessioned2026-07-07T05:26:13Z
dc.date.available2026-07-07T05:26:13Z
dc.descriptionMartin boundaries and integral representations of positive functions which are harmonic in a bounded domain $D$ with respect to Brownian motion are well understood. Unlike the Brownian case, there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process. One kind are functions harmonic in $D$ with respect to the whole process $X$, and the other are functions harmonic in $D$ with respect to the process $X^D$ killed upon leaving $D$. In this paper we show that for bounded Lipschitz domains, the Martin boundary with respect to the killed stable process $X^D$ can be identified with the Euclidean boundary. We further give integral representations for both kinds of positive harmonic functions. Also given is the conditional gauge theorem conditioned according to Martin kernels and the limiting behaviors of the $h$-conditional stable process, where $h$ is a positive harmonic function of $X^D$. In the case when $D$ is a bounded $C^{1, 1}$ domain, sharp estimate on the Martin kernel of $D$ is obtained.
dc.identifierhttps://arxiv.org/abs/math/9809179
dc.identifierhttp://arxiv.org/abs/math/9809179
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/77467
dc.subjectProbability
dc.titleMartin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes
dc.typetext

Files

Collections

arXiv