On Mean Field Convergence and Stationary Regime

Assume that a family of stochastic processes on some Polish space $E$ converges to a deterministic process; the convergence is in distribution (hence in probability) at every fixed point in time. This assumption holds for a large family of processes, among which many mean field interaction models and is weaker than previously assumed. We show that any limit point of an invariant probability of the stochastic process is an invariant probability of the deterministic process. The results are valid in discrete and in continuous time