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

TL;DR

This paper develops new techniques to analyze the behavior of symmetric random walks on groups with polynomial volume growth, especially under slowly varying moment conditions, providing sharp bounds on return probabilities.

Contribution

It introduces methods to study random walks driven by measures with slow moment growth, including a sharp lower bound for return probabilities with finite weak-logarithmic moments.

Findings

Established a sharp lower bound for return probabilities

Analyzed random walks with slowly varying moment conditions

Extended understanding of random walk behavior on polynomial volume growth groups

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.