The master equation for mean field game systems with fractional and nonlocal diffusions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915118475902976 |
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| author | Jakobsen, Espen Robstad Rutkowski, Artur |
| author_facet | Jakobsen, Espen Robstad Rutkowski, Artur |
| contents | 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 Lévy 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 Hölder spaces $C^γ_b$ 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 $N\to\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_18867 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The master equation for mean field game systems with fractional and nonlocal diffusions Jakobsen, Espen Robstad Rutkowski, Artur Analysis of PDEs 35Q89, 35S10, 35A01, 35A08, 35K08, 49L12, 45K05, 35K61, 46E15, 46G05 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 Lévy 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 Hölder spaces $C^γ_b$ 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 $N\to\infty$. |
| title | The master equation for mean field game systems with fractional and nonlocal diffusions |
| topic | Analysis of PDEs 35Q89, 35S10, 35A01, 35A08, 35K08, 49L12, 45K05, 35K61, 46E15, 46G05 |
| url | https://arxiv.org/abs/2305.18867 |