Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit

Samuel Herrmann (1, 2) , Julian Tugaut (3)

1 IECN - Institut Élie Cartan de Nancy
2 TOSCA - Simuler et calibrer des modèles stochastiques
3 BIGS - Biology, genetics and statistics

Self-stabilizing diffusions are stochastic processes, solutions of nonlinear stochastic differential equation, which are attracted by their own law. This specific self-interaction leads to singular phenomenons like non uniqueness of associated stationary measures when the diffusion leaves in some non convex environment. The aim of this paper is to describe these invariant measures and especially their asymptotic behavior as the noise intensity in the nonlinear SDE becomes small. We prove in particular that the limit measures are discrete measures and point out some properties of their support which permit in several situations to describe explicitly the whole set of limit measures. This study requires essentially generalized Laplace's method approximations.

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

Domaines: Probabilités [math.PR]

Samuel Herrmann, Julian Tugaut. Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit. Electronic Journal of Probability, 2010, 15, pp.2087-2116. ⟨10.1214/EJP.v15-842⟩. ⟨hal-00397470⟩