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 $\alpha(x)+1$, 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 $\alpha+\beta\geq2.$ If the function $x\longmapsto f_x$ is periodic and if the set $\{x:\alpha(x)=\alpha_0:=\inf_{x\in\R}\alpha(x)\}$ 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 $\alpha_0\geq1.$