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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2509.04647 |
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| _version_ | 1866918136057430016 |
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| author | Graber, P. Jameson Matter, Elizabeth Bolanos, Jesus Ruiz |
| author_facet | Graber, P. Jameson Matter, Elizabeth Bolanos, Jesus Ruiz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04647 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean Field Games of Controls with Fractional Laplacian Graber, P. Jameson Matter, Elizabeth Bolanos, Jesus Ruiz Analysis of PDEs 35Q89 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. |
| title | Mean Field Games of Controls with Fractional Laplacian |
| topic | Analysis of PDEs 35Q89 |
| url | https://arxiv.org/abs/2509.04647 |