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

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 $α(x)+1$, with $α(x)\in(0,2)$. If $f_x(y)$ is the density of a symmetric $α$-stable distribution for negative $x$ and the density of a symmetric $β$-stable distribution for non-negative $x$, where $α,β\in(0,2)$, then the chain is recurrent if and only if $α+β\geq2.$ If the function $x\longmapsto f_x$ is periodic and if the set $\{x:α(x)=α_0:=\inf_{x\in\R}α(x)\}$ 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 $α_0\geq1.$