# Two-scale homogenization of a stationary mean-field game

## TL;DR

This paper investigates the large-scale behavior of a stationary mean-field game with oscillating potentials using two-scale homogenization, deriving effective equations and proving solution existence and uniqueness.

## Contribution

It introduces a rigorous two-scale homogenization approach for stationary mean-field games with oscillating potentials, establishing effective macroscopic models.

## Key findings

- Derivation of two-scale homogenized MFG equations
- Proof of existence and uniqueness of solutions to the homogenized problems
- Characterization of the macroscopic behavior of oscillating MFGs

## Abstract

In this paper, we characterize the asymptotic behavior of a first-order stationary mean-field game (MFG) with a logarithm coupling, a quadratic Hamiltonian, and a periodically oscillating potential. This study falls into the realm of the homogenization theory, and our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems, which encode the so-called macroscopic or effective behavior of the original oscillating MFG. Moreover, we prove existence and uniqueness of the solution to these limit problems.