Convergence rates of self-repelling diffusions on Riemannian manifolds

TL;DR

This paper analyzes the convergence rates of self-repelling diffusions on compact Riemannian manifolds, revealing their relation to Ornstein-Uhlenbeck processes and providing bounds that improve upon previous results.

Contribution

It introduces a spectral decomposition approach to analyze self-repelling diffusions, deriving new upper and lower bounds on their convergence rates.

Findings

The diffusion is a second-order lift of an Ornstein-Uhlenbeck process.

Derived general upper bounds on convergence to stationarity.

Established lower bounds that improve previous results in the periodic case.

Abstract

We study a class of self-repelling diffusions on compact Riemannian manifolds whose drift is the gradient of a potential accumulated along their trajectory. When the interaction potential admits a suitable spectral decomposition, the dynamics and its environment are equivalent to a finite-dimensional degenerate diffusion. We show that this diffusion is a second-order lift of an Ornstein-Uhlenbeck process whose invariant law corresponds to the Gaussian invariant measure of the environment, and immediately obtain a general upper bound on the rate of convergence to stationarity using the framework of second-order lifts. Furthermore, using a flow Poincar\'e inequality, we develop lower bounds on the convergence rate. We show that, in the periodic case, these lower bounds improve upon those of Bena\"im and Gauthier (Probab. Theory Relat. Fields, 2016), and even match the order of the upper bound in some cases.