On Mean Field Games in Infinite Dimension
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916932571103232 |
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| author | Federico, Salvatore Gozzi, Fausto Święch, Andrzej |
| author_facet | Federico, Salvatore Gozzi, Fausto Święch, Andrzej |
| contents | We study a Mean Field Games (MFG) system in a real, separable infinite dimensional Hilbert space. The system consists of a second order parabolic type equation, called Hamilton-Jacobi-Bellman (HJB) equation in the paper, coupled with a nonlinear Fokker-Planck (FP) equation. Both equations contain a Kolmogorov operator. Solutions to the HJB equation are interpreted in the mild solution sense and solutions to the FP equation are interpreted in an appropriate weak sense. We prove well-posedness of the considered MFG system under certain conditions. The existence of a solution to the MFG system is proved using Tikhonov's fixed point theorem in a proper space. Uniqueness of solutions is obtained under typical separability and Lasry-Lions type monotonicity conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Mean Field Games in Infinite Dimension Federico, Salvatore Gozzi, Fausto Święch, Andrzej Analysis of PDEs Optimization and Control 35Q89, 35R15, 49N80, 35Q84, 49L12, 60H15, 91A16 We study a Mean Field Games (MFG) system in a real, separable infinite dimensional Hilbert space. The system consists of a second order parabolic type equation, called Hamilton-Jacobi-Bellman (HJB) equation in the paper, coupled with a nonlinear Fokker-Planck (FP) equation. Both equations contain a Kolmogorov operator. Solutions to the HJB equation are interpreted in the mild solution sense and solutions to the FP equation are interpreted in an appropriate weak sense. We prove well-posedness of the considered MFG system under certain conditions. The existence of a solution to the MFG system is proved using Tikhonov's fixed point theorem in a proper space. Uniqueness of solutions is obtained under typical separability and Lasry-Lions type monotonicity conditions. |
| title | On Mean Field Games in Infinite Dimension |
| topic | Analysis of PDEs Optimization and Control 35Q89, 35R15, 49N80, 35Q84, 49L12, 60H15, 91A16 |
| url | https://arxiv.org/abs/2411.14604 |