Weak and renormalized solutions to a hypoelliptic Mean Field Games system

TL;DR

This paper proves the existence and uniqueness of weak and renormalized solutions for a hypoelliptic Mean Field Games system, extending previous results to degenerate and quadratic growth Hamiltonians using kinetic regularity and De-Giorgi methods.

Contribution

It introduces new existence and uniqueness results for degenerate, hypoelliptic MFG systems with local coupling, including solutions with quadratic Hamiltonians.

Findings

Established $L^{ abla}-$bounds for degenerate Fokker-Planck equations.

Proved existence and uniqueness of weak solutions with Lipschitz Hamiltonians.

Proved existence and uniqueness of renormalized solutions with quadratic growth Hamiltonians.

Abstract

We establish the existence and uniqueness of weak and renormalized solutions to a degenerate, hypoelliptic Mean Field Games system with local coupling. An important step is to obtain $L^{\infty}-$bounds for solutions to a degenerate Fokker-Planck equation with a De-Giorgi type argument. In particular, we show existence and uniqueness of weak solutions to Mean Fields Games systems with Lipschitz Hamiltonians. Furthermore, we establish existence and uniqueness of renormalized solutions for Hamiltonians with quadratic growth. Our approach relies on the kinetic regularity of hypoelliptic equations obtained by Bouchut and the work of Porretta on the existence and uniqueness of renormalized solutions for the Mean Field Game system, in the non-degenerate setting.