Exit-time of mean-field particles system

Univ Lyon, Université Jean Monnet, CNRS UMR 5208, Institut Camille Jordan, Maison de l’Université, 10 rue Tréfilerie, CS 82301, 42023 Saint-Étienne Cedex 2, France.

^* Corresponding author: tugaut@math.cnrs.fr

Received: 27 July 2018
Accepted: 2 December 2019

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 ℝ^d 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 $\text{exp}\left\{\frac{2}{{\sigma }^2}{H}^N\right\}$, H^N being the exit-cost. We also show that this exit-cost converges to some quantity H.