Captivity of mean-fi eld particle systems and the related exit problems

Abstrakt

A mean-field system is a weakly interacting system of N particles in ℜ^d confined by an external potential. The aim of this work is to  establish a simple result about the exit problem of mean-field systems from some domains when the number of particles goes to infinity. More precisely, we prove the existence of some subsets of ℜ^dN such that the probability of leaving these sets before any T > 0 is arbitrarily small by taking N large enough. On the one hand, we show that the number of steady states in the small-noise limit is arbitrarily large with a sufficiently large number of particles. On the other hand, using the long-time convergence of the hydrodynamical limit, we identify the steady states as N goes to infinity with the invariant probabilities of the McKean–Vlasov diffusion so that some steady states in the small-noise limit are not steady states in the large N limit.