A strongly degenerate fully nonlinear mean field game with nonlocal diffusion

TL;DR

This paper establishes existence and uniqueness results for a novel class of fully nonlinear, degenerate mean field game systems involving nonlocal jump diffusions, expanding the theoretical understanding of such complex models.

Contribution

It introduces the first well-posedness results for degenerate, fully nonlinear MFGs with nonlocal diffusion and monotone couplings, using viscosity solutions and advanced analytical techniques.

Findings

Proved existence and uniqueness of solutions for the degenerate, fully nonlinear MFG system.

Developed a non-standard doubling of variables technique for degenerate non-smooth Fokker-Planck equations.

Extended the theory of MFGs to include nonlocal jump diffusions of order less than one.

Abstract

There are few results on mean field game (MFG) systems where the PDEs are either fully nonlinear or have degenerate diffusions. This paper introduces a problem that combines both difficulties. We prove existence and uniqueness for a strongly degenerate, fully nonlinear MFG system by using the well-posedness theory for fully nonlinear MFGs established in our previous paper. It is the first such application in a degenerate setting. Our MFG involves a controlled pure jump (nonlocal) L\'evy diffusion of order less than one, and monotone, smoothing couplings. The key difficulty is obtaining uniqueness for the corresponding degenerate, non-smooth Fokker-Plank equation: since the regularity of the coefficient and the order of the diffusion are interdependent, it holds when the order is sufficiently low. Viscosity solutions and a non-standard doubling of variables argument are used along with a bootstrapping procedure.