2014/08/15 by Pablo Lessa, Lessa, Pablo · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.DG #math.PR
paper · pdf · doi:10.48550/arxiv.1408.3571
arxiv created 2014/08/15 · openalex publication_date 2014/08/15 · arxiv updated 2014/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.