Un crit\`ere de r\'ecurrence pour certains espaces homog\`enes

TL;DR

This paper establishes a criterion for recurrence or transience of Markov chains on homogeneous spaces of semi-simple Lie groups, showing a uniform recurrence property under certain algebraic and probabilistic conditions.

Contribution

It provides a new recurrence criterion for Markov chains on homogeneous spaces of semi-simple Lie groups, demonstrating a uniform recurrence property across all points.

Findings

Either all trajectories are almost surely transient or recurrent.

Existence of a compact set that all trajectories visit infinitely often.

Recurrence criterion depends on the structure of G, H, and the measure μ.

Abstract

Let $G$ be a real connected algebraic semi-simple Lie group, and $H$ an algebraic subgroup of $G$. Let $\mu$ be a probability measure on $G$, with finite exponential moment, whose support spans a Zariski-dense subsemigroup of $G$. Let $X=G/H$ be the quotient of $G$ by $H$. We study the Markov chain on $X$ with transition probability $P_x=\mu *\delta_x$ for $x\in X$. We prove that either for every $x\in X$, almost every trajectory starting from $x$ is transient or for every $x\in X$, almost every trajectory starting from $x$ is recurrent. In fact, this recurrence is uniform over all $X$, i.e. there exists a compact set $C\subset X$ such that for each point $x\in X$, every trajectory starting in $x$ almost surely returns to $C$ infinitely often. Furthermore, we give a criterion for recurrence depending on $G$, $H$, and $\mu$.