Rate of convergence of some self-attracting diffusions

We consider the rate of convergence of the paths of some self-attracting diffusions. We prove that these diffusions are attracted by their mean-process. Moreover, under some assumptions on the interaction function f, this attraction becomes strong enough to imply the almost sure convergence of the paths. In this case, we provide an estimate of the rate of convergence which depends on the growth of f near the origin.