Long-Time Behavior of Deterministic Mean Field Games with Nonmonotone Interactions

We consider deterministic mean field games (MFGs) in all Euclidean space with a cost functional continuous with respect to the distribution of the agents and attaining its minima in a compact set. We first show that the static MFG with such a cost has an equilibrium, and we build from it a solution of the ergodic MFG system of first order PDEs with the same cost. Next we address the long-time limit of the solutions to finite horizon MFGs with cost functional satisfying various additional assumptions, but not the classical Lasry-Lions monotonicity condition. Instead we assume that the cost has the same set of minima for all measures describing the population. We prove the convergence of the distribution of the agents and of the value function to a solution of the ergodic MFG system as the horizon of the game tends to infinity, extending to this class of MFGs some results of weak KAM theory.