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arXiv · 2411.14604

On Mean Field Games in Infinite Dimension

Salvatore Federico·Fausto Gozzi·Andrzej Święch
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Abstract

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.

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Salvatore Federico, Fausto Gozzi, Andrzej Święch. 2024-11-21. On Mean Field Games in Infinite Dimension. https://arxiv.org/abs/2411.14604

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