Convergence in Distribution of Some Self-interacting Diffusions: the Simulated Annealing Method

We study some self-interacting diffusions living on R solutions to: dXt = dBt − g(t)∇V (Xt − μt)dt where μt is the empirical mean of the process X , V is an asymptotically strictly convex potential and g is a given function, not increasing too fast to the infinity or constant. The authors have already proved that the ergodic behavior of X is strongly related to g. We go further and, using the simulated annealing method, we give some conditions for the convergence in distribution of X toward X∞ (which law is related to the global minima of V ). We also investigate the case g(t) = 1.