On groups, slow heat kernel decay yields Liouville property and sharp entropy bounds

TL;DR

This paper links slow decay of heat kernels on groups to the Liouville property and provides sharp entropy bounds, extending previous results and applying to transitive graphs.

Contribution

It establishes new connections between heat kernel decay rates, the Liouville property, and entropy bounds on groups and graphs, improving earlier results.

Findings

Slow heat kernel decay implies the Liouville property.

Entropy of convolution powers is bounded by a power law.

Results are sharp and extend to transitive graphs.

Abstract

Let $\mu$ be a symmetric probability measure of finite entropy on a group $G$. We show that if $-\log \mu^{(2n)}(id)=o(n^{1/2})$, then the pair $(G,\mu)$ has the Liouville property (all bounded $\mu$-harmonic functions on $G$ are constant). Furthermore, if $-\log \mu^{(2n)}(id)=O(n^{\beta})$ where $\beta\in(0,1/2)$, then the entropy of the $n$-fold convolution power $\mu^{(n)}$ satisfies $H(\mu^{(n)})=O\left(n^{\frac{\beta}{1-\beta}}\right)$. This improves earlier results of Gournay and of Saloff-Coste and the second author. We extend the bounds to transitive graphs and illustrate their sharpness on a family of groups.