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

TL;DR

This paper explores qualitative properties and numerical simulations of ergodic Mean Field Games systems, focusing on the effective Hamiltonian and drift, and analyzing when the MFG structure is preserved or lost.

Contribution

It provides new qualitative insights into the effective Hamiltonian and drift in ergodic MFG systems, including conditions for structure preservation and numerical validation.

Findings

Effective Hamiltonian and drift properties are characterized.

Conditions identified for when MFG structure is preserved or lost.

Numerical simulations validate theoretical qualitative properties.

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 $\nabla_P \bar H(P,\alpha)-\bar b(P, \alpha)$.