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

A mean-field system is a weakly interacting system of N particles in ℜd 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 ℜdN 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.A mean-field system is a weakly interacting system of N particles in ℜd 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 ℜdN 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