Symmetric stable processes in cones
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.
scientific article; zbMATH DE number 2093719