arXiv subjects
Publications and source records attributed to Caroline Bruère.
We study the action of the affine group $G$ of $\mathbb{R}^d$ on the space $X_{k,\,d}$ of $k$-dimensional affine subspaces. Given a compactly-supported Zariski dense probability measure $μ$ on $G$, we show that $X_{k,\,d}$ supports a $μ$-stationary measure $ν$ if and only if the $(k\!+\!1)^{\rm th}$-Lyapunov exponent of $μ$ is strictly negative. In particular, when $μ$ is symmetric, $ν$ exists if and only if $2k\geq d$.
Take $G$ a locally compact second-countable group, and $H$ a subgroup of $G$. Choose $μ$ 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 $P_x=μ*δ_x$ for $x\in X$. Under some conditions on $G$, $H$, $μ$, 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.
Let $G$ be a real connected algebraic semi-simple Lie group, and $H$ an algebraic subgroup of $G$. Let $μ$ 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 $P_x=μ*δ_x$ 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 $μ$.