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 $p(x,\xi )=-i\beta (x)\xi +\gamma (x)\mathrm{\mid }\xi {\mathrm{\mid }}^{\alpha (x)},$ where $\alpha (x)\in (0,2)$, $\beta (x)\in \mathbb{R}$ and $\gamma (x)\in (0,\mathrm{\infty })$. 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 $\alpha \mathrm{\ne }1.$