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June  2012, 7(2): 279-301. doi: 10.3934/nhm.2012.7.279

Long time average of mean field games

Pierre Cardaliaguet 1, , Jean-Michel Lasry 2, , Pierre-Louis Lions 1, and  Alessio Porretta 3,

1. 

Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France, France
2. 

56 Rue d'Assas, 75006 Paris, France
3. 

Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scienti ca 1, 00133 Roma, Italy

Received  November 2011 Revised  March 2012 Published  June 2012

We consider a model of mean field games system defined on a time interval $[0,T]$ and investigate its asymptotic behavior as the horizon $T$ tends to infinity. We show that the system, rescaled in a suitable way, converges to a stationary ergodic mean field game. The convergence holds with exponential rate and relies on energy estimates and the Hamiltonian structure of the system.
Keywords: Mean Field Games, large time behavior, ergodic problem..
Mathematics Subject Classification: Primary: 35B40; Secondary: 35K5.
Citation: Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279
References:
[1]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM J. Numer. Anal., 48 (2010), 1136.  doi: 10.1137/090758477.  Google Scholar

[2]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem,, SIAM J. Control Opt., 50 (2012), 77.  doi: 10.1137/100790069.  Google Scholar

[3]

M. Arisawa and P.-L. Lions, On ergodic stochastic control,, Comm. Partial Differential Equations, 23 (1998), 2187.  doi: 10.1080/03605309808821413.  Google Scholar

[4]

G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations,, SIAM J. Math. Anal., 32 (2001), 1311.  doi: 10.1137/S0036141000369344.  Google Scholar

[5]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359.  doi: 10.1017/S0308210500018631.  Google Scholar

[6]

D. A. Gomes, J. Mohr and R. Souza, Discrete time, finite state space mean field games,, J. Math. Pures Appl. (9), 93 (2010), 308.   Google Scholar

[7]

D. A. Gomes, G. E. Pires and H. Sanchez-Morgado, A-priori estimates for stationary mean-field games,, preprint., ().   Google Scholar

[8]

D. A. Gomes and H. Sanchez-Morgado, A stochastic Evans-Aronsson problem,, preprint., ().   Google Scholar

[9]

O. Guéant, Mean field games with quadratic hamiltonian: A constructive scheme,, preprint., ().   Google Scholar

[10]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1967).   Google Scholar

[11]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Math. Acad. Sci. Paris, 343 (2006), 619.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[12]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal,, C. R. Math. Acad. Sci. Paris, 343 (2006), 679.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[13]

J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229.   Google Scholar

[14]

J.-M. Lasry, P.-L. Lions and O. Guéant, Application of mean field games to growth theory,, preprint, (2008).   Google Scholar

[15]

J.-M. Lasry and P.-L. Lions, Cours au Collège de France., Available from: \url{http://www.college-de-france.fr}., ().   Google Scholar

[16]

A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations,, Ann. Mat. Pura Appl. (4), 177 (1999), 143.  doi: 10.1007/BF02505907.  Google Scholar

show all references

References:
[1]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM J. Numer. Anal., 48 (2010), 1136.  doi: 10.1137/090758477.  Google Scholar

[2]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem,, SIAM J. Control Opt., 50 (2012), 77.  doi: 10.1137/100790069.  Google Scholar

[3]

M. Arisawa and P.-L. Lions, On ergodic stochastic control,, Comm. Partial Differential Equations, 23 (1998), 2187.  doi: 10.1080/03605309808821413.  Google Scholar

[4]

G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations,, SIAM J. Math. Anal., 32 (2001), 1311.  doi: 10.1137/S0036141000369344.  Google Scholar

[5]

L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE,, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359.  doi: 10.1017/S0308210500018631.  Google Scholar

[6]

D. A. Gomes, J. Mohr and R. Souza, Discrete time, finite state space mean field games,, J. Math. Pures Appl. (9), 93 (2010), 308.   Google Scholar

[7]

D. A. Gomes, G. E. Pires and H. Sanchez-Morgado, A-priori estimates for stationary mean-field games,, preprint., ().   Google Scholar

[8]

D. A. Gomes and H. Sanchez-Morgado, A stochastic Evans-Aronsson problem,, preprint., ().   Google Scholar

[9]

O. Guéant, Mean field games with quadratic hamiltonian: A constructive scheme,, preprint., ().   Google Scholar

[10]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1967).   Google Scholar

[11]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Math. Acad. Sci. Paris, 343 (2006), 619.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[12]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal,, C. R. Math. Acad. Sci. Paris, 343 (2006), 679.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[13]

J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229.   Google Scholar

[14]

J.-M. Lasry, P.-L. Lions and O. Guéant, Application of mean field games to growth theory,, preprint, (2008).   Google Scholar

[15]

J.-M. Lasry and P.-L. Lions, Cours au Collège de France., Available from: \url{http://www.college-de-france.fr}., ().   Google Scholar

[16]

A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations,, Ann. Mat. Pura Appl. (4), 177 (1999), 143.  doi: 10.1007/BF02505907.  Google Scholar

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