Degenerate Mean Field Games with H\"ormander diffusion

TL;DR

This paper investigates degenerate mean field game systems with H"ormander diffusion, establishing existence, uniqueness, and regularity of solutions despite degeneracy challenges.

Contribution

It proves the existence and uniqueness of classical solutions for degenerate mean field games with H"ormander diffusion using novel regularity techniques.

Findings

Established global Schauder estimates for degenerate PDE systems.

Proved the weak maximum principle via subsolution construction.

Extended regularity results to degenerate operators with local homogeneity.

Abstract

In this paper, we study a class of degenerate mean field game systems arising from the mean field games with H\"ormander diffusion, where the generic player may have a ``forbidden'' direction at some point. Here we prove the existence and uniqueness of the classical solutions in weighted H\"older spaces for the PDE systems, which describe the Nash equilibria in the games. The degeneracy causes the lack of commutation of vector fields and the fundamental solution which are the main difficulties in the proof of the global Schauder estimate and the weak maximum principle. Based on the idea of the localizing technique and the local homogeneity of degenerate operators, we extend the maximum regularity result and obtain the global Schauder estimates. For the weak maximum principle, we construct a subsolution instead of the fundamental solution of the degenerate operators.