On the long time behavior of potential MFG.

Summary This thesis deals with the study of the long time behavior of potential mean field games (MFG), independently of the convexity of the associated minimization problem. For the finite dimensional Hamiltonian system, similar problems have been treated by the weak KAM theory. We transpose many results of this theory to the context of potential mean field games. First, we characterize by ergodic approximation the boundary value associated with finite horizon MFG systems. We provide explicit examples in which this value is strictly greater than the energy level of the stationary solutions of the ergodic MFG system. This implies that the optimal trajectories of finite horizon MFG systems cannot converge to stationary configurations. Then, we prove the convergence of the minimization problem associated with finite horizon MFG to a solution of the critical Hamilton-Jacobi equation in the space of probability measures. Moreover, we show a mean field limit for the ergodic constant associated to the corresponding finite dimensional Hamilton-Jacobi equation. In the last part, we characterize the limit of the infinite horizon minimization problem that we used for the ergodic approximation in the first part of the manuscript.