\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A-priori estimates for stationary mean-field games

Abstract Related Papers Cited by
  • 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 $u$.
    Mathematics Subject Classification: 35J47, 49L25, 49N70.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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. TranAdjoint methods for obstacle problems and weakly coupled systems of PDE, submitted.

    [5]

    F. Cagnetti, D. Gomes and H. V. TranAubry-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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(208) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return