Recurrence and transience criteria for two cases of stable-like Markov chains

Abstract

We give recurrence and transience criteria for two cases of time-homogeneous Markov chains on the real line with transition kernel $p(x,dy)=f_x(y-x)dy$, where $f_x(y)$ are probability densities of symmetric distributions and, for large $|y|$, have a power-law decay with exponent $\alpha (x)+1$, with $\alpha (x)\in (0,2)$. If $f_x(y)$ is the density of a symmetric $\alpha$-stable distribution for negative $x$ and the density of a symmetric $\beta$-stable distribution for non-negative $x$, where $\alpha ,\beta \in (0,2)$, then the chain is recurrent if and only if $\alpha +\beta \ge 2.$ If the function $x⟼f_x$ is periodic and if the set $\{x:\alpha (x)={\alpha }_0:={\inf }_{x\in \mathbb{R}}\alpha (x)\}$ has positive Lebesgue measure, then, under a uniformity condition on the densities $f_x(y)$ and some mild technical conditions, the chain is recurrent if and only if ${\alpha }_0\ge 1.$

References in corpus (1)

• Correction to: The scaling limit behaviour of periodic stable-like processes

Cited by in corpus (1)

• Continuous-time open quantum walks in one dimension: matrix-valued orthogonal polynomials and Lindblad generators