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 $ar H$ and of the effective drift $ar b$. In particular we shall provide two cases where the effective system keeps/looses the Mean Field Games structure, namely where $ abla_P ar H(P,alpha)$ coincides or not with $ar b(P, alpha)$. par On the other hand, we shall provide some numerical tests validating the aforementioned qualitative properties of $ar H$ and $ar b$. In particular, we provide a numerical estimate of the discrepancy $ abla_P ar H(P,alpha)-ar b(P, alpha)$.