Captivity of mean-field systems∗

The aim of this work is to establish the stability of mean-field system under non-convex confining potential. A mean-field system corresponds to a system of N particles in weak interaction and confined by an exterior force. With our hypotheses, it is a Kolmogorov diffusion with potential Υ N . Exit time of these systems have been studied in details in the smallnoise limit. Here, we will deal with the large-dimension limit with fixed noise. In one hand, we show that the meta-potential Υ admits a number of wells which tends to infinity when N goes to infinity. In the other hand, by using the convergence of McKean-Vlasov processes in long-time and the propagation of chaos, we prove that there exist traps such that the diffusion can not escape from. Furthermore, the traps do not coincide with the wells of Υ .