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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.00152 |
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| _version_ | 1866912009947185152 |
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| author | Chowdhury, Indranil Jakobsen, Espen R. Krupski, Miłosz |
| author_facet | Chowdhury, Indranil Jakobsen, Espen R. Krupski, Miłosz |
| contents | There are few results on mean field game (MFG) systems where the PDEs are either fully nonlinear or have degenerate diffusions. This paper introduces a problem that combines both difficulties. We prove existence and uniqueness for a strongly degenerate, fully nonlinear MFG system by using the well-posedness theory for fully nonlinear MFGs established in our previous paper. It is the first such application in a degenerate setting. Our MFG involves a controlled pure jump (nonlocal) Lévy diffusion of order less than one, and monotone, smoothing couplings. The key difficulty is obtaining uniqueness for the corresponding degenerate, non-smooth Fokker-Plank equation: since the regularity of the coefficient and the order of the diffusion are interdependent, it holds when the order is sufficiently low. Viscosity solutions and a non-standard doubling of variables argument are used along with a bootstrapping procedure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00152 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A strongly degenerate fully nonlinear mean field game with nonlocal diffusion Chowdhury, Indranil Jakobsen, Espen R. Krupski, Miłosz Analysis of PDEs 35A01, 35A02, 35D30, 35D40, 35K55, 35K65, 35Q84, 35Q89, 35R11, 47D07, 49L, 49N80, 60G51 There are few results on mean field game (MFG) systems where the PDEs are either fully nonlinear or have degenerate diffusions. This paper introduces a problem that combines both difficulties. We prove existence and uniqueness for a strongly degenerate, fully nonlinear MFG system by using the well-posedness theory for fully nonlinear MFGs established in our previous paper. It is the first such application in a degenerate setting. Our MFG involves a controlled pure jump (nonlocal) Lévy diffusion of order less than one, and monotone, smoothing couplings. The key difficulty is obtaining uniqueness for the corresponding degenerate, non-smooth Fokker-Plank equation: since the regularity of the coefficient and the order of the diffusion are interdependent, it holds when the order is sufficiently low. Viscosity solutions and a non-standard doubling of variables argument are used along with a bootstrapping procedure. |
| title | A strongly degenerate fully nonlinear mean field game with nonlocal diffusion |
| topic | Analysis of PDEs 35A01, 35A02, 35D30, 35D40, 35K55, 35K65, 35Q84, 35Q89, 35R11, 47D07, 49L, 49N80, 60G51 |
| url | https://arxiv.org/abs/2409.00152 |