Self-Interacting Diffusions : Symmetric Interactions

Let $M$ be a compact Riemannian manifold. A {\em self-interacting diffusion} on $M$ is a stochastic process solution to $$d{X}_t=d{W}_t(X_t)-\frac{1}{t}({\int }_0^t\mathrm{\nabla }{V}_{X_s}(X_t)ds)dt$$ where $\{W_t\}$ is a Brownian vector field on $M$ and $V_x(y)=V(x,y)$ a smooth function. Let $\mu_t = \frac{1}{t} \int_0^t \delta_{X_s} ds$ denote the normalized occupation measure of $X_t$. We prove that, when $V$ is symmetric, ${\mu }_t$ converges almost surely to the critical set of a certain nonlinear free energy functional $J$. Furthermore, $J$ has generically finitely many critical points and ${\mu }_t$ converges almost surely toward a local minimum of $J\mathrm{.}$ Each local minimum having a positive probability to be selected.