Homogenization of a mean field game system in the small noise limit

TL;DR

This paper investigates how homogenization and small noise limits affect second-order mean field game systems, showing convergence to an effective first-order system with properties derived from a cell problem, often losing the original MFG structure.

Contribution

It demonstrates the combined effect of homogenization and small noise limits on second-order MFG systems, introducing an effective first-order system derived from a cell problem.

Findings

Solutions converge to an effective first-order system

Effective operators are characterized through a cell problem

The effective system generally loses the MFG structure

Abstract

This paper concerns the simultaneous effect of homogenization and of the small noise limit for a $2^{\textrm {nd}}$ order mean field games (MFG) system with local coupling and quadratic Hamiltonian. We show under some additional assumptions that the solutions of our system converge to a solution of an effective $1^{\textrm {st}}$ order system whose effective operators are defined through a cell problem which is a $2^{\textrm {nd}}$ order system of ergodic MFG type. We provide several properties of the effective operators and we show that in general the effective system looses the MFG structure.