Long time average of mean field games with a nonlocal coupling

We study the long time average, as the time horizon tends to infinity, of the solution of a mean field game system with a nonlocal coupling. We show an exponential convergence to the solution of the associated stationary ergodic mean field game. Proofs rely on semiconcavity estimates and smoothing properties of the linearized system. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdA