Linear drift and entropy for regular covers

TL;DR

This paper investigates the relationship between linear drift and entropy in Brownian motion on regular Riemannian covers, establishing an inequality linking these geometric invariants.

Contribution

It proves that the square of the linear drift is less than or equal to the Kaimanovich entropy for regular Riemannian covers.

Findings

Established the inequality  \,  h for geometric invariants.

Provides insights into the asymptotic behavior of Brownian motion.

Connects geometric properties with probabilistic invariants.

Abstract

We consider a regular Riemannian cover $\M$ of a compact Riemannian manifold. The linear drift $\ell$ and the Kaimanovich entropy $h$ are geometric invariants defined by asymptotic properties of the Brownian motion on $\M$. We show that $\ell^2 \leq h$.