Ergodic property of stable-like Markov chains

TLDR: In this paper, under a certain uniformity condition on the density functions and additional mild drift conditions, the authors gave sufficient conditions for recurrence and transience for a symmetric stable random walk on the real line.

Abstract: A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the transition kernel $$p(x,\hbox {d}y)=f_x(y-x)\hbox {d}y$$ , 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 $$\alpha \ne 1$$ .