Long-time behavior of stable-like processes

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 $\alpha\neq1.$