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 of the corresponding semigroup given by Lf(x)=a(x)f′(x)+∫Rf(x+u)-f(x)-∇f(x)·u1|u|≤1ν(x,du),f∈C∞2(R).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Lf(x)= & {} a(x)f'(x)\\{} & {} + \int _\mathbb {R}\left( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb {1}_{|u|\le 1} \right) \nu (x,du), \quad f\in C_\infty ^2(\mathbb {R}). \end{aligned}$$\end{document}Here ν(x,du)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu (x,du)$$\end{document} is a Lévy-type kernel and a(·):R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(\cdot ): \mathbb {R}\rightarrow \mathbb {R}$$\end{document}. We consider the case where the tails of ν(x,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu (x,\cdot )$$\end{document} have polynomial decay, as well as the case where the decay is (sub)-exponential. We use the Foster–Lyapunov approach to prove the results.