On ergodic properties of some Levy-type processes

In this note we prove some sufficient conditions for ergodicity of a Levy-type process, such that on the test functions the generator of the respective semigroup is of the form $$Lf(x)=a(x)f^{\mathrm{\prime }}(x)+{\int }_{\mathbb{R}}(f(x+u)-f(x)-\mathrm{\nabla }f(x)\cdot u{\mathbb{I}}_{\mathrm{\mid }u\mathrm{\mid }\le 1})\nu (x,du),f\in C_{\mathrm{\infty }}^2(\mathbb{R})\mathrm{.}$$ Here $\nu (x,du)$ is a Levy-type kernel and $a(\cdot ):\mathbb{R}\to \mathbb{R}$. We consider the case when the tails are of polynomial decay as well as the case when the decay is (sub)-exponential. For the proof the Foster-Lyapunov approach is used.