Symmetric stable processes in cones
type: Item
instance of: scholarly article
title: Symmetric stable processes in cones
zbMATH Open document ID: 1054.31002
author: Rodrigo Bañuelos
author: Krzysztof Bogdan
published in: Potential Analysis
publication date: 2004-08-20T00:00:00Z
review text: The main result identifies those \(p > 0\) for which \(E_ {x} \tau _ {\Gamma }^ {p} < \infty \). \(\tau _ {\Gamma }\) denotes the first exit time, from a generalized cone \(\Gamma \), of a \(\alpha\)-stable rotation invariant Lévy process \(X_ t\) in \({\mathbb R}^ d\). By a generalized cone is meant an open set \(\Gamma \subset {\mathbb R}^ d\) such that \(x\in \Gamma \) and \(r > 0\) \(\Rightarrow \) \(r x \in \Gamma \). The same problem for the process killed off the cone and conditioned by the Martin kernel with pole at \(0\) is also considered. These results are analogues of those obtained by \textit{R. Bañuelos} and \textit{R. G. Smits} [Probab. Theory Relat. Fields 108, No. 3, 299--319 (1997; Zbl 0884.60037)] and by \textit{B. Davis} and \textit{B. Zhang} [Proc. Am. Math. Soc. 121, No. 3, 925--929 (1994; Zbl 0805.60079)]. Similar results, for right circular cones, were obtained by \textit{R. D. DeBlassie} [Ann. Probab. 18, No. 3, 1034--1070 (1990; Zbl 0709.60075)] and by \textit{P. J. Méndez-Hernández} [Ill. J. Math. 46, No. 1, 155--163 (2002; Zbl 1011.60023)]. During the study, the Martin kernel with pole at infinity or at \(0\) is constructed and its uniqueness is proved. Also, the spherical fractional Laplacian is studied.
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 2093719
zbMATH Keywords: symmetric stable process
zbMATH Keywords: \(\alpha\)-harmonic function
zbMATH Keywords: cone
zbMATH Keywords: conditional process
zbMATH Keywords: exit time
Wikidata QID: Q110598167
reviewed by: Liliana Popa
MaRDI profile type: Publication
DOI: 10.1023/B:POTA.0000033333.72236.DC
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description: scientific article; zbMATH DE number 2093719