A Monotone--Operator Proof of Existence and Uniqueness for a Simple Stationary Mean Field Game

TL;DR

This paper provides an accessible, detailed proof of existence and uniqueness for a simple stationary mean field game using a monotone-operator approach, suitable for motivated undergraduates.

Contribution

It offers a clear, elementary exposition of the monotone-operator method applied to a specific mean field game, including a detailed framework and explicit Hamiltonian case.

Findings

Existence of a strong solution established

Uniqueness of the solution proven

Method applicable to similar mean field games

Abstract

We study a stationary first--order mean field game on the $d$--dimensional torus. The system couples a Hamilton--Jacobi equation for the value function with a transport equation for the density of players. Our goal is to give a detailed and friendly exposition of the monotone--operator argument that yields existence and uniqueness of solutions. We first present a general framework in a Hilbert space and prove existence of a strong solution by adding a simple coercive regularisation and applying Minty's method. Then we specialise to the explicit Hamiltonian \[ H(p,m)=|p|^2-m, \] check all assumptions, and show how the abstract theorem gives existence and uniqueness for this concrete mean field game. The exposition is written in a slow and elementary way so that a motivated undergraduate can follow each step.