Homogenization of a Mean Field Game System in the Small Noise Limit

This paper concerns the simultaneous effect of homogenization and of the small noise limit for a second order mean field game (MFG) system with local coupling and quadratic Hamiltonian. We show under some additional assumptions that the solutions of our system converge to a solution of an effective first order system whose effective operators are defined through a cell problem which is a second order system of ergodic MFG type. We provide several properties of the effective operators, and we show that in general the effective system loses the MFG structure.

[1]  P. Lions,et al.  Mean field games , 2007 .

[2]  P. Lions,et al.  Jeux à champ moyen. I – Le cas stationnaire , 2006 .

[3]  Pierre Cardaliaguet,et al.  Weak Solutions for First Order Mean Field Games with Local Coupling , 2013, 1305.7015.

[4]  Diogo A. Gomes,et al.  Mean Field Games Models—A Brief Survey , 2013, Dynamic Games and Applications.

[5]  D. Lukkassen,et al.  On Weak Convergence of Locally Periodic Functions , 2002, math/0210293.

[6]  D. Lacker.  A general characterization of the mean field limit for stochastic differential games , 2014, 1408.2708.

[7]  Martino Bardi,et al.  Nonlinear elliptic systems and mean-field games , 2016 .

[8]  M. Burger,et al.  Mean field games with nonlinear mobilities in pedestrian dynamics , 2013, 1304.5201.

[9]  Markus Fischer.  On the connection between symmetric $N$-player games and mean field games , 2014, 1405.1345.

[10]  Diogo A. Gomes,et al.  Time dependent mean-field games in the superquadratic case , 2013 .

[11]  Fabio Camilli,et al.  Stationary Mean Field Games Systems Defined on Networks , 2015, SIAM J. Control. Optim..

[12]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[13]  Pierre-Louis Lions,et al.  Long time average of mean field games , 2012, Networks Heterog. Media.

[14]  Roland P. Malhamé,et al.  A micro-macro traffic model based on Mean-Field Games , 2015, 2015 American Control Conference (ACC).

[15]  A. Bensoussan.  Perturbation Methods in Optimal Control , 1988 .

[16]  Pierre-Louis Lions,et al.  Long Time Average of Mean Field Games with a Nonlocal Coupling , 2013, SIAM J. Control. Optim..

[17]  Marco Cirant,et al.  Stationary focusing mean-field games , 2016, 1602.04231.

[18]  Yves Achdou,et al.  Finite Difference Methods for Mean Field Games , 2013 .

[19]  P. Lions,et al.  Jeux à champ moyen. II – Horizon fini et contrôle optimal , 2006 .

[20]  L. Cesari,et al.  Contributions to modern calculus of variations , 1987 .

[21]  Pierre Cardaliaguet,et al.  Mean field games systems of first order , 2014, 1401.1789.

[22]  Diogo A. Gomes,et al.  A-priori estimates for stationary mean-field games , 2012, Networks Heterog. Media.

[23]  A. Bensoussan,et al.  Mean Field Games and Mean Field Type Control Theory , 2013 .

[24]  Marie-Therese Wolfram,et al.  On a mean field game approach modeling congestion and aversion in pedestrian crowds , 2011 .

[25]  Grigorios A. Pavliotis,et al.  Multiscale Methods: Averaging and Homogenization , 2008 .