Self-interacting diffusions IV: Rate of convergence

TL;DR

This paper investigates the rate at which self-interacting diffusions on compact manifolds converge to their limiting measure, establishing a central limit theorem that quantifies how interaction strength influences convergence speed.

Contribution

It provides a rigorous analysis of the convergence rate for self-interacting diffusions and proves a central limit theorem describing the fluctuations around the limit.

Findings

Convergence rate is faster with stronger repelling interactions.

A central limit theorem characterizes the fluctuations of the empirical measure.

The results extend understanding of long-term behavior of self-interacting diffusions.

Abstract

Self-interacting diffusions are processes living on a compact Riemannian manifold defined by a stochastic differential equation with a drift term depending on the past empirical measure of the process. The asymptotics of this measure is governed by a deterministic dynamical system and under certain conditions it converges almost surely towards a deterministic measure (see Bena\"im, Ledoux, Raimond (2002) and Bena\"im, Raimond (2005)). We are interested here in the rate of this convergence. A central limit theorem is proved. In particular, this shows that greater is the interaction repelling faster is the convergence.