  The present 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 empirical mean of
the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is
a given function. The authors have still studied the ergodic behavior of $X$
and proved that it is strongly related to $g$. We go further and give necessary
and sufficient conditions (for small $g$'s) in order that $X$ converges in
probability to $X_\infty$ (which is related to the global minima of $V$).
