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arXiv · 2409.00152

A strongly degenerate fully nonlinear mean field game with nonlocal diffusion

Indranil Chowdhury·Espen R. Jakobsen·Miłosz Krupski
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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.

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Indranil Chowdhury, Espen R. Jakobsen, Miłosz Krupski. 2024-08-30. A strongly degenerate fully nonlinear mean field game with nonlocal diffusion. https://arxiv.org/abs/2409.00152

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