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Minimax Theory and its Applications 03 (2018), No. 2, 211--226
Copyright Heldermann Verlag 2018

An Ergodic Problem for Mean Field Games: Qualitative Properties and Numerical Simulations

Simone Cacace
Dip. di Matematica e Fisica, UniversitÓ degli Studi Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy

Fabio Camilli
Dip. di Scienze di Base e Applicate per l'Ingegneria, UniversitÓ di Roma "Sapienza", Via Scarpa 16, 00161 Roma, Italy

Annalisa Cesaroni
Dip. di Scienze Statistiche, UniversitÓ di Padova, Via Cesare Battisti 241-243, 35121 Padova, Italy

Claudio Marchi
Dip. di Ingegneria dell'Informazione, UniversitÓ di Padova, Via Gradenigo 6/B, 35131 Padova, Italy


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 $\nabla_P \bar H(P,\alpha)$ coincides or not with $\bar b(P, \alpha)$. \par 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 $\nabla_P \bar H(P,\alpha)-\bar b(P, \alpha)$.

Keywords: Mean field games, periodic homogenization, small noise limit, ergodic problems, continuous dependence of solution on parameters, finite difference schemes.

MSC: 35B27, 35B30, 35B40, 35K40, 35K59, 65M06, 91A13.

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