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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 itssupport 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 $x\in X$. Undersome conditions on $G$, $H$, $\mu$, we know that this Markov chain is eithereverywhere recurrent or everywhere transient. A natural question is whethersuch a dichotomy is universally true. The goal of this paper is to show it isnot, even when $G$ is finitely generated, through the explicit construction ofa counter example. The methods used include the study of stable laws,Gnedenko-Kolmogorov's local limit theorem for stable laws, and the study of thetime of first return to equilibrium.

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