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

Long time average of mean field games

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.
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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[1]

Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173

[2]

Juan Pablo Maldonado López. Discrete time mean field games: The short-stage limit. Journal of Dynamics & Games, 2015, 2 (1) : 89-101. doi: 10.3934/jdg.2015.2.89

[3]

Ahmed Bonfoh, Cyril D. Enyi. Large time behavior of a conserved phase-field system. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1077-1105. doi: 10.3934/cpaa.2016.15.1077

[4]

Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311

[5]

Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks & Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021

[6]

Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics & Games, 2019, 0 (0) : 1-19. doi: 10.3934/jdg.2019016

[7]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243

[8]

Yves Achdou, Victor Perez. Iterative strategies for solving linearized discrete mean field games systems. Networks & Heterogeneous Media, 2012, 7 (2) : 197-217. doi: 10.3934/nhm.2012.7.197

[9]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303

[10]

Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks & Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315

[11]

Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465

[12]

Toyohiko Aiki, Adrian Muntean. Large time behavior of solutions to a moving-interface problem modeling concrete carbonation. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1117-1129. doi: 10.3934/cpaa.2010.9.1117

[13]

Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096

[14]

Geonho Lee, Sangdong Kim, Young-Sam Kwon. Large time behavior for the full compressible magnetohydrodynamic flows. Communications on Pure & Applied Analysis, 2012, 11 (3) : 959-971. doi: 10.3934/cpaa.2012.11.959

[15]

Diogo Gomes, Marc Sedjro. One-dimensional, forward-forward mean-field games with congestion. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 901-914. doi: 10.3934/dcdss.2018054

[16]

Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control & Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026

[17]

Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks & Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263

[18]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[19]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557

[20]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

[Back to Top]