Brownian motion on stationary random manifolds

TL;DR

This paper introduces stationary random manifolds, develops their entropy theory, and characterizes when they have zero or positive entropy based on harmonic functions and geometric properties.

Contribution

It defines stationary random manifolds and establishes a fundamental entropy theory, linking entropy to harmonic functions and geometric growth.

Findings

Entropy is zero iff the manifold satisfies the Liouville property almost surely.

Entropy is positive iff there exists an infinite dimensional space of bounded harmonic functions.

Bounds for entropy are given in terms of linear drift and volume growth.

Abstract

We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volume growth of the manifold. Other almost sure properties of these random manifolds are also studied.