Convergence in distribution of some self-interacting diffusions

The present paper is concerned with some self-interacting diffusions (xt,t ≥ 0) living on d. these diffusions are solutions to stochastic differential equations: dxt = dbt - g (t) δ v (xt - μ ̄t) dt,where μ̄ 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 ∫ xe- 2v(x) dx = 0. © 2014 aline kurtzmann.