Convergence in distribution of some particular self-interacting diffusions: the simulated annealing method

Abstract

The present paper is concerned with some self-interacting diffusions $(X_t,t\ge 0)$ living on ${\mathbb{R}}^d$. These diffusions are solutions to stochastic differential equations: $$\mathrm{d}{X}_t=\mathrm{d}{B}_t-g(t)\nabla V(X_t-{\bar{\mu }}_t)\mathrm{d}t$$ where ${\bar{\mu }}_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. The authors have still studied the ergodic behavior of $X$ and proved that it is strongly related to $g$. We go further and give necessary and sufficient conditions (for small $g$'s) in order that $X$ converges in probability to $X_{\infty }$ (which is related to the global minima of $V$).

companion paper to 0707.2908