A Monotone--Operator Proof of Existence and Uniqueness for a Simple Stationary Mean Field Game
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915666368397312 |
|---|---|
| author | Ismatov, Hikmatullo |
| author_facet | Ismatov, Hikmatullo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09671 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Monotone--Operator Proof of Existence and Uniqueness for a Simple Stationary Mean Field Game Ismatov, Hikmatullo Functional Analysis 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. |
| title | A Monotone--Operator Proof of Existence and Uniqueness for a Simple Stationary Mean Field Game |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2512.09671 |