Symmetric stable processes stay in thick sets.
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.
scientific article; zbMATH DE number 2100720
scientific article