Self-stabilizing processes in multi-wells landscape in -convergence

Self-stabilizing processes are inhomogeneous diffusions in which the law of the process intervenes in the drift. If the external force is the gradient of a convex potential, it has been proved that the process converges towards the unique invariant probability as the time goes to infinity. However, in a previous article, we established that the diffusion may admit several invariant probabilities, provided that the external force derives from a non-convex potential. We here provide results about the limiting values of the family { μ t ; t ≥ 0 } , μ t being the law of the diffusion. Moreover, we establish the weak convergence under an additional hypothesis.