Symmetric stable processes stay in thick sets.
type: Item
instance of: scholarly article
title: Symmetric stable processes stay in thick sets.
zbMATH Open document ID: 1054.60057
published in: The Annals of Probability
publication date: 2004-09-15T00:00:00Z
review text: Let \(W(f)=\{0<x_1<1,\,x_2^2+\cdots+ x_d^2<f^2(x_1)\}\) be a thorn in \(\mathbb R^d\) for some nondecreasing and left continuous function \(f:(0,1)\to(0,\infty)\) with \(f(0+)=0\). For symmetric \(\alpha\)-stable processes \(X(t)\) in \(\mathbb R^d\) with \(0<\alpha<2\), \textit{K. Burdzy} and \textit{T. Kulczycki} [Ann. Probab. 31, No. 1, 170--194 (2003; Zbl 1019.60035)] give an exact integral condition on \(f\) under which there exists a random time \(S\) such that \(X[S,S+1)\) stays in \(X(S)+\overline{W(f)}\) with probability either 1 or 0. The present paper extends this result to arbitrary open sets \(W\) in \(\mathbb R^d\) with \(0\in\partial W\). Due to the jump nature of the process, \(W\) does not even need to be locally connected at 0. The authors give sufficient conditions which tell how thick or how thin \(W\) has to be such that the corresponding above probability is 1 or 0. The first is an integral condition on the expected exit time of \(W\), the latter is a rather technical condition on certain partitions of \(W\). Three examples of lacunary rings, blocks of varying shape and scattered cubes are given in which the conditions are sharp in the sense that the probability can only be 0 or 1.
Mathematics Subject Classification ID: 60G52
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 31C45
zbMATH DE Number: 2100720
zbMATH Keywords: symmetric stable process
zbMATH Keywords: thick sets
author: Jang-Mei Wu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1078415837
OpenAlex ID: W2008393136
cites work: Q4888858
cites work: On the Distribution of First Hits for the Symmetric Stable Processes
cites work: Brownian paths and cones
cites work: Stable processes have thorns
cites work: On some relations between the harmonic measure and the Levy measure for a certain class of Markov processes
DOI: 10.1214/AOP/1078415837
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description: scientific article; zbMATH DE number 2100720