Ergodic property of stable-like Markov chains

A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the transition kernel $p(x,dy)=f_x(y-x)dy$, where the density functions $f_x(y)$, for large $|y|$, have a power-law decay with exponent $\alpha(x)+1$, where $\alpha(x)\in(0,2)$. In this paper, under a certain uniformity condition on the density functions $f_x(y)$ and additional mild drift conditions, we give sufficient conditions for recurrence in the case when $0<\liminf_{|x|\longrightarrow\infty}\alpha(x)$, sufficient conditions for transience in the case when $\limsup_{|x|\longrightarrow\infty}\alpha(x)<2$ and sufficient conditions for ergodicity in the case when $0<\inf\{\alpha(x):x\in\mathbb{R}\}$. As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric $\alpha$-stable random walk on $\mathbb{R}$ with the index of stability $\alpha\neq1.$