Non uniqueness of stationary measures for self-stabilizing diffusions

Samuel Herrmann (1) , Julian Tugaut (1)

1 IECN - Institut Élie Cartan de Nancy

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 to point out singular phenomenons 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.

Mots clés: fixed point theorem; self-interacting diffusion; stationary measures; double well potential; perturbed dynamical system; Laplace's method; fixed point theorem.

Domaines: Probabilités [math.PR]

Samuel Herrmann, Julian Tugaut. Non uniqueness of stationary measures for self-stabilizing diffusions. Stochastic Processes and their Applications, 2010, 120 (7), pp.1215-1246. ⟨hal-00368177⟩