The ODE method for some self-interacting diffusions on ${ℝ}^d$

Kurtzmann, Aline

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 618-643.

Le but de cet article est d'étudier le comportement asymptotique d'une classe de processus en auto-interaction sur ℝd. Ces processus de diffusion s'écrivent comme solution d'E.D.S. dont le terme de dérive dépend à la fois de la position actuelle du processus et de sa mesure empirique μt. Jusqu'à présent, Benaïm, Ledoux et Raimond ont mené l'étude de ce type de diffusions sur des espaces compacts, via des méthodes d'approximation stochastique. Nous étendons ces techniques à ℝd, en supposant l'existence d'un potentiel de confinement (vérifiant certaines conditions). Nous avons besoin de ces hypothèses sur le potentiel de confinement, car, en général, un tel processus peut être transient. Nous concluons cet article par un exemple sur ℝ2.

The aim of this paper is to study the long-term behavior of a class of self-interacting diffusion processes on ℝd. These are solutions to SDEs with a drift term depending on the actual position of the process and its normalized occupation measure μt. These processes have so far been studied on compact spaces by Benaïm, Ledoux and Raimond, using stochastic approximation methods. We extend these methods to ℝd, assuming a confinement potential satisfying some conditions. These hypotheses on the confinement potential are required since in general the process can be transient, and is thus very difficult to analyze. Finally, we illustrate our study with an example on ℝ2.

Classification : 60K35, 37C50
Mots clés : self-interaction diffusion, reinforced processes, stochastic approximation

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