Random walks under slowly varying moment conditions on groups of polynomial volume growth

Saloff-Coste, Laurent; Zheng, Tianyi

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Numéro Spécial : Conférence “Talking Across Fields” du 24 au 28 mars 2014 à l’Institut de Mathématiques de Toulouse, Volume 24 (2015) no. 4, pp. 837-855.

Let $G$ be a finitely generated group of polynomial volume growth equipped with a word-length $|·|$. The goal of this paper is to develop techniques to study the behavior of random walks driven by symmetric measures $\mu$ such that, for any $ϵ>0$, ${\sum |·|}^{ϵ}\mu =\infty$. In particular, we provide a sharp lower bound for the return probability in the case when $\mu$ has a finite weak-logarithmic moment.

Soit $G$ un groupe finiment engendré, à croissance polynomiale du volume et muni de la distance des mots associée à un ensemble donné de générateurs. Le but de ce travail est de développer des techniques qui permettent l’étude de marches aléatoires associées à des mesures de probabilité symetriques, $\mu$, telles que, pout tout $ϵ>0$, ${\sum |·|}^{ϵ}\mu =\infty$. En particulier, nous donnons une borne inférieure optimale pour la probabilité de retour dans le cas où $\mu$ a un moment logarithmique de type faible fini.

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