Long-time behavior of stable-like processes

TL;DR

This paper investigates the long-term behavior of stable-like processes, providing conditions for recurrence, transience, and ergodicity based on their defining functions, and offers new insights into symmetric stable Lévy processes.

Contribution

It establishes new sufficient conditions for recurrence, transience, and ergodicity of stable-like processes, including a novel proof for symmetric stable Lévy processes.

Findings

Conditions for recurrence and transience based on process parameters

New proof for recurrence/transience of symmetric stable Lévy processes

Characterization of ergodicity in terms of stability and drift functions

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 $p(x,\xi)=-i\beta(x)\xi+\gamma(x)|\xi|^{\alpha(x)},$ where $\alpha(x)\in(0,2)$, $\beta(x)\in\R$ and $\gamma(x)\in(0,\infty)$. More precisely, we give sufficient conditions for recurrence, transience and ergodicity of stable-like processes in terms of the stability function $\alpha(x)$, the drift function $\beta(x)$ and the scaling function $\gamma(x)$. Further, as a special case of these results we give a new proof for the recurrence and transience property of one-dimensional symmetric stable L\'{e}vy processes with the index of stability $\alpha\neq1.$