Un contre-exemple \`a la dichotomie r\'ecurrence/transience sur les espaces homog\`enes

Abstract

Take $G$ a locally compact second-countable group, and $H$ a subgroup of $G$. Choose $\mu$ a probability measure on $G$, such that the group spanned by its support is dense in $G$, and consider the Markov chain on the homogeneous space $X=G/H$ with transition probability $P_x=\mu *\delta_x$ for $x\in X$. Under some conditions on $G$, $H$, $\mu$, we know that this Markov chain is either everywhere recurrent or everywhere transient. A natural question is whether such a dichotomy is universally true. The goal of this paper is to show it is not, even when $G$ is finitely generated, through the explicit construction of a counter example. The methods used include the study of stable laws, Gnedenko-Kolmogorov's local limit theorem for stable laws, and the study of the time of first return to equilibrium.