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January 2004 Symmetric stable processes stay in thick sets
Jang-Mei Wu
Ann. Probab. 32(1A): 315-336 (January 2004). DOI: 10.1214/aop/1078415837

Abstract

Let $X(t)$ be the symmetric $\alpha$-stable process in $\Bbb R^d (0<\alpha<2, d\ge 2)$. Then let $W(f)$ be the thorn $\{x \in \Bbb R^d\dvtx 0<x_1<1, (x_2^2 +\cdots + x_d^2)^{1/2} < f(x_1) \}$ where $f\dvtx (0,1)\rightarrow(0,1)$ is continuous, increasing with $f(0^+) = 0$. Recently Burdzy and Kulczycki gave an exact integral condition on f for the existence of a random time s such that $X(t)$ remains in the thorn $X(s)+ \overline{W(f)}$ for all $t \in [s,s+1)$. We extend their theorem to general open sets W with $0 \in \partial W$. In general, $\alpha$-processes may stay in sets which are quite lacunary and are not locally connected at 0.

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Jang-Mei Wu. "Symmetric stable processes stay in thick sets." Ann. Probab. 32 (1A) 315 - 336, January 2004. https://doi.org/10.1214/aop/1078415837

Information

Published: January 2004
First available in Project Euclid: 4 March 2004

zbMATH: 1054.60057
MathSciNet: MR2040784
Digital Object Identifier: 10.1214/aop/1078415837

Subjects:
Primary: 60G17, 60G52
Secondary: 31C45

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 1A • January 2004
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