Wu, Jang-Mei Symmetric stable processes stay in thick sets. (English) Zbl 1054.60057 Ann. Probab. 32, No. 1A, 315-336 (2004). 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 \(0<\alpha<2\), K. Burdzy and 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. Reviewer: Peter Becker-Kern (Debrecen) Cited in 1 Document MSC: 60G52 Stable stochastic processes 60G17 Sample path properties 31C45 Other generalizations (nonlinear potential theory, etc.) Keywords:symmetric stable process; thick sets Citations:Zbl 1019.60035 PDFBibTeX XMLCite \textit{J.-M. Wu}, Ann. Probab. 32, No. 1A, 315--336 (2004; Zbl 1054.60057) Full Text: DOI References: [1] Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press. · Zbl 0861.60003 [2] Blumenthal, R. M., Getoor, R. K. and Ray, D. B. (1961). On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99 540–554. · Zbl 0118.13005 [3] Burdzy, K. (1985). Brownian paths and cones. Ann. Probab. 13 1006–1010. JSTOR: · Zbl 0574.60053 [4] Burdzy, K. and Kulczycki, T. (2003). Stable processes have thorns. Ann. Probab. 31 170–194. · Zbl 1019.60035 [5] Ikeda, N. and Watanabe, S. (1962). On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2 79–95. · Zbl 0118.13401 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.