An ergodic problem for mean field games: qualitative properties and numerical simulations

This paper is devoted to some qualitative descriptions and some numerical results for ergodic Mean Field Games systems which arise, e.g., in the homogenization with a small noise limit. We shall consider either power type potentials or logarithmic type ones. In both cases, we shall establish some qualitative properties of the effective Hamiltonian H¯ and of the effective drift b¯. In particular we shall provide two cases where the effective system keeps/looses the Mean Field Games structure, namely where ∇PH¯(P,α) coincides or not with b¯(P,α). On the other hand, we shall provide some numerical tests validating the aforementioned qualitative properties of H¯ and b¯. In particular, we provide a numerical estimate of the discrepancy ∇PH¯(P,α)−b¯(P,α).