A quadratic Mean Field Games model for the Langevin equation

TL;DR

This paper develops a Mean Field Games model with Langevin dynamics and quadratic costs, transforming it into coupled kinetic Fokker-Planck equations and proving the existence of solutions.

Contribution

It introduces a novel approach by transforming the MFG system into coupled kinetic Fokker-Planck equations and proves the existence of solutions for this system.

Findings

Existence of solutions for the coupled kinetic Fokker-Planck equations.

Transformation of the MFG system into a kinetic Fokker-Planck system.

Validation of the model's well-posedness.

Abstract

We consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. A change of variables, introduced in [9], transforms the Mean Field Games system into a system of two coupled kinetic Fokker-Planck equations. We prove an existence result for the latter system, obtaining consequently existence of a solution for the Mean Field Games system.