Non-uniqueness of stationary measures for self-stabilizing processes

We investigate the existence of invariant measures for self-stabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a double-well landscape and attracted by its own law. This specific self-interaction leads to nonlinear stochastic differential equations and permits pointing out singular phenomena like non-uniqueness of associated stationary measures. The existence of several invariant measures is essentially based on the non-convex environment and requires generalized Laplace’s method approximations.