Long-time behavior of stable-like processes

Abstract

In this paper, we consider a long-time behavior of stable-like processes. A stable-like process is a Feller process given by the symbol $p(x,ξ)=-iβ(x)ξ+γ(x)|ξ|^{α(x)},$ where $α(x)\in(0,2)$, $β(x)\in\R$ and $γ(x)\in(0,\infty)$. More precisely, we give sufficient conditions for recurrence, transience and ergodicity of stable-like processes in terms of the stability function $α(x)$, the drift function $β(x)$ and the scaling function $γ(x)$. Further, as a special case of these results we give a new proof for the recurrence and transience property of one-dimensional symmetric stable Lévy processes with the index of stability $α\neq1.$