Mean Field Games of Controls with Fractional Laplacian

P. Jameson Graber; Elizabeth Matter; Jesus Ruiz Bolanos

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arXiv · 2509.04647

Mean Field Games of Controls with Fractional Laplacian

P. Jameson Graber·Elizabeth Matter·Jesus Ruiz Bolanos

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Abstract

We analyze a fractional mean field game of controls system, showing existence of solutions when the order of the fractional Laplacian is $s\in(\frac{1}{2},1)$. Here the running cost depends on the distribution $μ$ of not only the states but also optimal strategies. The coupling is assumed to satisfy the Lasry-Lions monotonicity condition. We derive three types of a priori estimates on solutions. First, we use the monotonicity condition to derive moment estimates on $μ$. Second, we derive abstract estimates on fractional parabolic equations and apply them to the mean field game. Third, we derive new estimates on the time regularity of the distribution $μ$ by analyzing the associated Lévy process. We apply these estimates and the Leray-Schauder fixed point theorem to establish existence of solutions.

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BibTeX RIS

P. Jameson Graber, Elizabeth Matter, Jesus Ruiz Bolanos. 2025-09-04. Mean Field Games of Controls with Fractional Laplacian. https://arxiv.org/abs/2509.04647

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