June  2012, 7(2): 303-314. doi: 10.3934/nhm.2012.7.303

A-priori estimates for stationary mean-field games

1. 

Departamento de Matemática and CAMGSD, IST Avenida Rovisco Pais, Lisboa, Portugal, Portugal

2. 

Instituto de Matem, Universidad Nacional Aut, M, Mexico

Received  November 2011 Revised  March 2012 Published  June 2012

In this paper we establish a new class of a-priori estimates for stationary mean-field games which have a quasi-variational structure. In particular we prove $W^{1,2}$ estimates for the value function $u$ and that the players distribution $m$ satisfies $\sqrt{m}\in W^{1,2}$. We discuss further results for power-like nonlinearities and prove higher regularity if the space dimension is 2. In particular we also obtain in this last case $W^{2,p}$ estimates for $u$.
Citation: Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks and Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303
References:
[1]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477.

[2]

Julien Salomon, Aimée Lachapelle and Gabriel Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588. doi: 10.1142/S0218202510004349.

[3]

F. Camilli, Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Opt., 50 (2012), 77-109. doi: 10.1137/100790069.

[4]

F. Cagnetti, D. Gomes and H. V. Tran, Adjoint methods for obstacle problems and weakly coupled systems of PDE,, submitted., (). 

[5]

F. Cagnetti, D. Gomes and H. V. Tran, Aubry-Mather measures in the non convex setting,, submitted., (). 

[6]

Lawrence C. Evans and Charles K. Smart, Adjoint methods for the infinity Laplacian partial differential equation, Arch. Ration. Mech. Anal., 201 (2011), 87-113. doi: 10.1007/s00205-011-0399-x.

[7]

Lawrence C. Evans, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations, 17 (2003), 159-177. doi: 10.1007/s00526-002-0164-y.

[8]

Lawrence C. Evans, Further PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations, 35 (2009), 435-462. doi: 10.1007/s00526-008-0214-1.

[9]

L. C. Evans, Adjoint and compensated compactness methods for Hamilton-Jacobi PDE, Arch. Ration. Mech. Anal., 197 (2010), 1053-1088. doi: 10.1007/s00205-010-0307-9.

[10]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652. doi: 10.1016/S0764-4442(97)84777-5.

[11]

A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046.

[12]

A. Fathi, Orbite hétéroclines et ensemble de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213-1216.

[13]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.

[14]

D. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308-328.

[15]

D. Gomes, J. Mohr and R. R. Souza, Mean-field limit of a continuous time finite state game, preprint, 2011.

[16]

D. Gomes, A stochastic analogue of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304.

[17]

D. Gomes and H Sanchez-Morgado, On the stochastic Evans-Aronsson problem, preprint, 2011.

[18]

O. Gueant, "Mean Field Games and Applications to Economics," Ph.D. Thesis, Université Paris Dauphine, Paris, 2009.

[19]

O. Gueant, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), 276-294.

[20]

Minyi Huang, Peter E. Caines and Roland P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571. doi: 10.1109/TAC.2007.904450.

[21]

Minyi Huang, Roland P. Malhamé and Peter E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.

[22]

Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.

[23]

Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.

[24]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.

[25]

Jean-Michel Lasry and Pierre-Louis Lions, "Mean Field Games," Cahiers de la Chaire Finance et Développement Durable, 2007.

[26]

Jean-Michel Lasry, Pierre-Louis Lions and O. Guéant, Application of mean field games to growth theory, preprint, 2010.

[27]

Jean-Michel Lasry, Pierre-Louis Lions and O. Guéant, Mean field games and applications, in "Paris-Princeton Lectures on Mathematical Finance 2010," Lecture Notes in Math., 2003, Springer, Berlin, (2011), 205-266.

[28]

J. Mather, Action minimizing invariant measure for positive definite Lagrangian systems, Math. Z, 207 (1991), 169-207. doi: 10.1007/BF02571383.

[29]

Ricardo Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638.

[30]

Kaizhi Wang, Action minimizing stochastic invariant measures for a class of Lagrangian systems, Commun. Pure Appl. Anal., 7 (2008), 1211-1223.

show all references

References:
[1]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477.

[2]

Julien Salomon, Aimée Lachapelle and Gabriel Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588. doi: 10.1142/S0218202510004349.

[3]

F. Camilli, Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Opt., 50 (2012), 77-109. doi: 10.1137/100790069.

[4]

F. Cagnetti, D. Gomes and H. V. Tran, Adjoint methods for obstacle problems and weakly coupled systems of PDE,, submitted., (). 

[5]

F. Cagnetti, D. Gomes and H. V. Tran, Aubry-Mather measures in the non convex setting,, submitted., (). 

[6]

Lawrence C. Evans and Charles K. Smart, Adjoint methods for the infinity Laplacian partial differential equation, Arch. Ration. Mech. Anal., 201 (2011), 87-113. doi: 10.1007/s00205-011-0399-x.

[7]

Lawrence C. Evans, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations, 17 (2003), 159-177. doi: 10.1007/s00526-002-0164-y.

[8]

Lawrence C. Evans, Further PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations, 35 (2009), 435-462. doi: 10.1007/s00526-008-0214-1.

[9]

L. C. Evans, Adjoint and compensated compactness methods for Hamilton-Jacobi PDE, Arch. Ration. Mech. Anal., 197 (2010), 1053-1088. doi: 10.1007/s00205-010-0307-9.

[10]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652. doi: 10.1016/S0764-4442(97)84777-5.

[11]

A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046.

[12]

A. Fathi, Orbite hétéroclines et ensemble de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1213-1216.

[13]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.

[14]

D. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308-328.

[15]

D. Gomes, J. Mohr and R. R. Souza, Mean-field limit of a continuous time finite state game, preprint, 2011.

[16]

D. Gomes, A stochastic analogue of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304.

[17]

D. Gomes and H Sanchez-Morgado, On the stochastic Evans-Aronsson problem, preprint, 2011.

[18]

O. Gueant, "Mean Field Games and Applications to Economics," Ph.D. Thesis, Université Paris Dauphine, Paris, 2009.

[19]

O. Gueant, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), 276-294.

[20]

Minyi Huang, Peter E. Caines and Roland P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571. doi: 10.1109/TAC.2007.904450.

[21]

Minyi Huang, Roland P. Malhamé and Peter E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.

[22]

Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.

[23]

Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.

[24]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.

[25]

Jean-Michel Lasry and Pierre-Louis Lions, "Mean Field Games," Cahiers de la Chaire Finance et Développement Durable, 2007.

[26]

Jean-Michel Lasry, Pierre-Louis Lions and O. Guéant, Application of mean field games to growth theory, preprint, 2010.

[27]

Jean-Michel Lasry, Pierre-Louis Lions and O. Guéant, Mean field games and applications, in "Paris-Princeton Lectures on Mathematical Finance 2010," Lecture Notes in Math., 2003, Springer, Berlin, (2011), 205-266.

[28]

J. Mather, Action minimizing invariant measure for positive definite Lagrangian systems, Math. Z, 207 (1991), 169-207. doi: 10.1007/BF02571383.

[29]

Ricardo Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638.

[30]

Kaizhi Wang, Action minimizing stochastic invariant measures for a class of Lagrangian systems, Commun. Pure Appl. Anal., 7 (2008), 1211-1223.

[1]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457

[2]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583

[3]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383

[4]

Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423

[5]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial and Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165

[6]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial and Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363

[7]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101

[8]

Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277

[9]

Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1791-1801. doi: 10.3934/dcdss.2020105

[10]

Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control and Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365

[11]

Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523

[12]

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

[13]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics and Games, 2021, 8 (4) : 467-486. doi: 10.3934/jdg.2021014

[14]

Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks and Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279

[15]

Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics and Games, 2019, 6 (3) : 221-239. doi: 10.3934/jdg.2019016

[16]

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

[17]

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

[18]

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

[19]

Adriano Festa, Diogo Gomes, Francisco J. Silva, Daniela Tonon. Preface: Mean field games: New trends and applications. Journal of Dynamics and Games, 2021, 8 (4) : i-ii. doi: 10.3934/jdg.2021025

[20]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics and Games, 2021, 8 (4) : 445-465. doi: 10.3934/jdg.2021006

2020 Impact Factor: 1.213

Metrics

  • PDF downloads (140)
  • HTML views (0)
  • Cited by (28)

[Back to Top]