Ergodicity of self-attracting motion

TL;DR

This paper investigates the long-term behavior of self-attracting stochastic processes in Euclidean space, establishing conditions for their ergodicity and convergence speed by combining stochastic approximation and free energy methods.

Contribution

It introduces new sufficient conditions for the ergodicity of self-attracting diffusions, linking stochastic approximation with McKean-Vlasov processes.

Findings

Established conditions for ergodicity of self-attracting motions.

Derived convergence speed estimates for these processes.

Connected asymptotic behavior to free energy decrease in McKean-Vlasov models.

Abstract

The aim of this paper is to study the asymptotic behaviour of a class of self- attracting motions on R^d . Using stochastic approximation methods, these processes have already been studied by Bena\"im, Ledoux and Raimond (2002) in a compact setting. We also relate the asymptotic behaviour of the self-attracting Brownian motion to the McKean-Vlasov process that was studied, via the decrease of the free energy, by Carrillo, McCann and Villani (2003). Mixing these methods, we manage to obtain sufficient conditions for the (limit-quotient) ergodicity of the self-attracting diffusion, together with a speed of convergence.