On Ergodic Properties of Some Lévy-Type Processes

In this note we find sufficient conditions for ergodicity of a Lévy-type process with the generator corresponding semigroup given by $$\begin{aligned} Lf(x)= & {} a(x)f'(x)\\{} + \int _\mathbb {R}\left( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb {1}_{|u|\le 1} \right) \nu (x,du), \quad f\in C_\infty ^2(\mathbb {R}). \end{aligned}$$ Here $$\nu (x,du)$$ is kernel and $$a(\cdot ): {R}\rightarrow {R}$$ . We consider case where tails (x,\cdot )$$ have polynomial decay, as well decay (sub)-exponential. use Foster–Lyapunov approach to prove results.