Abstract
The current article is devoted to the study of a mean-field system of particles. The question that we are interested in is the behaviour of the exit-time of the first particle (and the one of any particle) from a domain D on ℝd as the diffusion coefficient goes to 0. We establish a Kramers’ type law. In other words, we show that the exit-time is exponentially equivalent to [see formula in PDF], HN being the exit-cost. We also show that this exit-cost converges to some quantity H.
Subject
Statistics and Probability
Reference7 articles.
1. Dembo A. and
Zeitouni O., Stochastic Modelling and Applied Probability, in Large deviations techniques and applications, Vol. 38.
Corrected reprint of the second (1998) edn.,
Springer-Verlag,
Berlin
(2010).
2. Freidlin M.I. and
Wentzell A.D., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], in Random perturbations of dynamical systems, Vol. 260. 2nd edn.,
Translated fromthe 1979 Russian original by Joseph Szücs.
Springer-Verlag,
New York
(1998).
3. Large deviations and a Kramers’ type law for self-stabilizing diffusions
4. Méléard S., Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, in Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), Vol. 1627 of
Lecture Notes in Mathematics,
Springer,
Berlin
(1996) 42–95.