Asymptotic Properties of Random Homology Induced by Diffusion Processes

TL;DR

This paper studies the long-term behavior of random homology from diffusion processes on manifolds, revealing a rigidity result linking quadratic rates to the manifold's fiber bundle structure and contrasting relaxation speeds of reversible and non-reversible processes.

Contribution

It establishes a novel rigidity theorem connecting quadratic rates to manifold structure and compares the relaxation behaviors of reversible and non-reversible diffusion processes.

Findings

Quadratic rate implies the manifold is a fiber bundle over a flat torus.

Non-reversible processes relax to equilibrium slower than reversible ones.

Homology relaxation behavior differs from empirical measure relaxation.

Abstract

We investigate the asymptotic behavior, in the long time limit, of the random homology associated to realizations of stochastic diffusion processes on a compact Riemannian manifold. In particular a rigidity result is established: if the rate is quadratic, then the manifold is a locally trivial fiber bundle over a flat torus, with fibers being minimal in a weighted sense (that is, regarding the manifold as a metric measured space, with the invariant probability being the weight measure). Surprisingly, this entails that at least for some classes of manifolds, the homology of non-reversible processes relaxes to equilibrium slower than its reversible counterpart (as opposed to the respective empirical measure, which relaxes faster).