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

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.

💡 Research Summary

The paper addresses a stationary first‑order mean‑field game (MFG) posed on the d‑dimensional flat torus (\mathbb T^{d}). The unknowns are a scalar value function (u:\mathbb T^{d}\to\mathbb R) and a probability density (m:\mathbb T^{d}\to