The master equation for mean field game systems with fractional and nonlocal diffusions

TL;DR

This paper establishes existence and uniqueness of solutions for the master equation in mean field games involving fractional and nonlocal diffusions, covering a broad class of Levy diffusions and including mixed local-nonlocal cases.

Contribution

It provides the first comprehensive well-posedness results for the master equation with fractional and nonlocal diffusions, including new auxiliary results for related PDEs.

Findings

Proved existence and uniqueness of classical solutions for the master equation.

Extended well-posedness results to mixed local-nonlocal diffusion operators.

Developed auxiliary results for viscous Hamilton-Jacobi and linear parabolic equations.

Abstract

We prove existence and uniqueness of classical solutions of the master equation for mean field game (MFG) systems with fractional and nonlocal diffusions. We cover a large class of L\'evy diffusions of order greater than one, including purely nonlocal, local, and even mixed local-nonlocal operators. In the process we prove refined well-posedness results for the MFG systems, results that include the mixed local-nonlocal case. We also show various auxiliary results on viscous Hamilton-Jacobi equations, linear parabolic equations, and linear forward-backward systems that may be of independent interest. This includes a rigorous treatment of certain equations and systems with data and solutions in the duals of H\"older spaces $C^\gamma_b$ on the whole of $\mathbb{R}^d$. We do not assume existence of any moments for the initial distributions of players. In a future work we will use the results of this paper to prove the convergence of $N$-player games to mean field games as $N\to\infty$.