Un critère de récurrence pour certains espaces homogènes
Un critère de récurrence pour certains espaces homogènes
摘要：
设$G$是实连通代数半单李群，$H$是$G$的代数子群.设$μ$是$G$上的概率测度，具有有限指数矩，其支持度跨越$G$的Zurski稠密子半群.令$X=G/H$为$G$与$H$的商。本文研究了$X$上的马尔可夫链，其转移概率为$x\in X$的$P_x=μ*δ_x$.我们证明了，对于每个$x\in X$，几乎每个从$x$开始的轨迹都是瞬态的，或者对于每个$x\in X$，几乎每个从$x$开始的轨迹都是递归的.事实上，这种递归在所有$X$上是一致的，即存在一个紧集$C\subset X$，使得对于每个点$x\in X$，从$x$开始的每个轨迹几乎必然无限频繁地返回到$C$。此外，我们给出了依赖于$G$，$H$和$μ$的复发准则。
英文摘要：
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 $μ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
