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

Abstract

This paper deals with some self-interacting diffusions $(X_t,t\ge 0)$ living on ${\mathbb{R}}^d$. These diffusions are solutions to stochastic differential equations: \[\mathrm{d}X_t=\mathrm{d}B_t-g(t)\nabla V(X_t-\barμ_t)\,\mathrm{d}t,\] 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. 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 ${\bar{\mu }}_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$.

Published in at http://dx.doi.org/10.3150/10-BEJ310 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• A survey of random processes with reinforcement
• An asymptotic result for Brownian polymers

Cited by in corpus (3)

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• Current fluctuations of a self-interacting diffusion on a ring
• Self-interacting processes via Doob conditioning