American Institute of Mathematical Sciences

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

A-priori estimates for stationary mean-field games

Diogo A. Gomes 1, , Gabriel E. Pires 1, and  Héctor Sánchez-Morgado 2,

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 $u$.
Keywords: a-priori estimates, Mean field games, variational and quasi-variational methods..
Mathematics Subject Classification: 35J47, 49L25, 49N7.
Citation: 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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

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