Convergence in Distribution of Some Self-Interacting Diffusions

Abstract EN

The present paper is concerned with some self-interacting diffusions ( X t , t ≥ 0 ) living on ℝ d .

These diffusions are solutions to stochastic differential equations: d X t = d B t - g ( t ) ∇ V ( X t - μ ¯ t ) d t , where μ ¯ t is the empirical mean of the process X , V is an asymptotically strictly convex potential, and g is a given positive function.

We study the asymptotic behaviour of X for three different families of functions g .

If g t = k log t with k small enough, then the process X converges in distribution towards the global minima of V , whereas if t g ( t ) → c ∈ ] 0 , + ∞ ] or if g ( t ) → g ( ∞ ) ∈ [ 0 , + ∞ [ , then X converges in distribution if and only if ∫ x e - 2 V ( x ) d x = 0 .