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arXiv · 2305.18867

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

Espen Robstad Jakobsen·Artur Rutkowski
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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 $N\to\infty$.

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Espen Robstad Jakobsen, Artur Rutkowski. 2023-05-30. The master equation for mean field game systems with fractional and nonlocal diffusions. https://arxiv.org/abs/2305.18867

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