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arXiv · 1801.08828

An ergodic problem for Mean Field Games: qualitative properties and numerical simulations

Simone Cacace·Fabio Camilli·Annalisa Cesaroni·Claudio Marchi
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Abstract

This paper is devoted to some qualitative descriptions and some numerical results for ergodic Mean Field Games systems which arise, e.g., in the homogenization with a small noise limit. We shall consider either power type potentials or logarithmic type ones. In both cases, we shall establish some qualitative properties of the effective Hamiltonian $\bar H$ and of the effective drift $\bar b$. In particular we shall provide two cases where the effective system keeps/looses the Mean Field Games structure, namely where $\nabla_P \bar H(P,α)$ coincides or not with $\bar b(P, α)$. On the other hand, we shall provide some numerical tests validating the aforementioned qualitative properties of $\bar H$ and $\bar b$. In particular, we provide a numerical estimate of the discrepancy $\nabla_P \bar H(P,α)-\bar b(P, α)$.

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Simone Cacace, Fabio Camilli, Annalisa Cesaroni, Claudio Marchi. 2018-01-26. An ergodic problem for Mean Field Games: qualitative properties and numerical simulations. https://arxiv.org/abs/1801.08828

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