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

Abstract

Let $G$ be a finitely generated group of polynomial volume growth equipped with a word-length $|\cdot |$. 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 $\epsilon >0$, $\sum |\cdot {|}^{\epsilon }\mu =\infty$. In particular, we provide a sharp lower bound for the return probability in the case when $\mu$ has a finite weak-logarithmic moment.