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

Abstract

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