Some particular self-interacting diffusions: ergodic behavior and almost sure convergence

This paper is concerned with some self-interacting diffusions $(X_t,t\geq 0)$ living on $\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t - g(t)\nabla V(X_t - \bar{\mu}_t) \mathrm{d}t,$$ where $\bar{\mu}_t$ is the mean of the empirical measure of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behavior of $X$ and prove that it is strongly related to $g$. Actually, we will show that $X$ and $\bar{\mu}_t$ have the same asymptotic behavior and we will give necessary and sufficient conditions (on $g$ and $V$) for the almost sure convergence of $X$.