Ergodicity of self-attracting motion

Victor A. Kleptsyn (1) , Aline Kurtzmann (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IECN - Institut Élie Cartan de Nancy

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 Benaïm, 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.

Domaines: Probabilités [math.PR]

Victor A. Kleptsyn, Aline Kurtzmann. Ergodicity of self-attracting motion. Electronic Journal of Probability, 2012, 17 (50), 37 pp. ⟨10.1214/EJP.v17-2121⟩. ⟨hal-00400810v2⟩