An ergodic problem for Mean Field Games: qualitative properties and numerical simulations

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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 ∇ₚ H(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 ∇ₚ H(P,α)- b(P, α).