Ergodicity of self-attracting motion

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.