A-priori estimates for stationary mean-field games

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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

Abstract
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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 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.

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