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

TL;DR

This paper constructs a counterexample to the universal recurrence/transience dichotomy for Markov chains on homogeneous spaces, challenging previous assumptions and expanding understanding of stochastic processes on groups.

Contribution

It provides the first explicit counterexample demonstrating that the recurrence-transience dichotomy does not always hold for Markov chains on homogeneous spaces.

Findings

Counterexample disproves the universal dichotomy

Uses stable laws and local limit theorems in construction

Analyzes first return times to equilibrium

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.