On the turnpike property for mean field games

We consider the behavior of mean field games systems in the long horizon, under the assumption of monotonicity of the coupling term. Assuming that the Hamiltonian is globally Lipschitz and locally uniformly convex, we show that the time dependent solution is exponentially close to the ergodic stationary state in the long intermediate stages. This is evidence of the so-called exponential turnpike property for optimal control problems. Indeed, our proof follows a general approach which relies on the stabilization through the Riccati feedback of the associated linearized system.