Convergence in Distribution of Some Self-Interacting Diffusions - Aline Kurtzmann (Journal of Probability and Statistics)
The present paper is concerned with some self-interacting diffusions (Xt,t≥0) living on ℝd. These diffusions are solutions to stochastic differential equations: dXt=dBt-g(t)∇V(Xt-μ¯t)dt, 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 gt=klog⁡t with k small enough, then the process X converges in distribution towards the global minima of V, whereas if tg(t)→c∈]0,+∞] or if g(t)→g(∞)∈[0,+∞[, then X converges in distribution if and only if∫xe-2V(x)  dx=0.