Self-interacting diffusions. III. Symmetric interactions

Let M be a compact Riemannian manifold. A self-interacting diffusion on M is a stochastic process solution to $$dX_{t}=dW_{t}(X_{t})-\frac{1}{t}\biggl(\int_{0}^{t}\nabla V_{X_{s}}(X_{t})\,ds\biggr)\,dt,$$ where {Wt} is a Brownian vector field on M and Vx(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 Xt. We prove that, when V is symmetric, μt converges almost surely to the critical set of a certain nonlinear free energy functional J. Furthermore, J has generically finitely many critical points and μt converges almost surely toward a local minimum of J. Each local minimum has a positive probability to be selected.