Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence

Abstract

This paper deals with some self-interacting diffusions $(X_t, t \geq 0)$ living on $ℝ^d$. These diffusions are solutions to stochastic differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t − g(t)∇V(X_t − \overline μ_t) \mathrm{d}t,$$ where $\overline μ_t$ is the empirical mean of the process $X, V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behaviour of $X$ and prove that it is strongly related to $g$. Actually, we show that $X$ is ergodic (in the limit quotient sense) if and only if $\overline μ_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$.