CAPTIVITY OF MEAN-FIELD PARTICLE SYSTEMS AND THE RELATED
EXIT PROBLEMS
Abstract: A mean-field system is a weakly interacting system of particles in 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
such that the probability of leaving these sets before any is arbitrarily
small by taking 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 hydro-
dynamical limit, we identify the steady states as 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 limit.
2000 AMS Mathematics Subject Classification: Primary: 82C22, 60F10; Secondary:
60J60, 60G10.
Keywords and phrases: Interacting particle system, propagation of chaos, exit time,
nonconvexity, free energy, invariant probabilities.