Long time average of mean field games

We consider a model of mean field games system defined on a time interval [0, T] and investigate its asymptotic behavior as the horizon T tends to infinity. We show that the system, rescaled in a suitable way, converges to a stationary ergodic mean field game. The convergence holds with exponential rate and relies on energy estimates and the Hamiltonian structure of the system. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdA