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A class of infinite horizon mean field games on networks

  • * Corresponding author: Yves Achdou

    * Corresponding author: Yves Achdou 

The authors were partially supported by ANR project ANR-16-CE40-0015-01. The work of O. Ley and N. Tchou was partially supported by the Centre Henri Lebesgue ANR-11-LABX-0020-01

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  • We consider stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the invariant measure $ m $, a value function $ u $, and the ergodic constant $ \rho $. The function $ u $ is continuous and satisfies general Kirchhoff conditions at the vertices. The invariant measure $ m $ satisfies dual transmission conditions: in particular, $ m $ is discontinuous across the vertices in general, and the values of $ m $ on each side of the vertices satisfy special compatibility conditions. Existence and uniqueness are proven under suitable assumptions.

    Mathematics Subject Classification: Primary: 35R02, 49N70; Secondary: 91A13.


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  • Figure 1.  Left: the network $ \Gamma $ in which the edges are oriented toward the vertex with larger index ($ 4 $ vertices and $ 4 $ edges). Right: a new network $ \tilde \Gamma $ obtained by adding an artificial vertex ($ 5 $ vertices and $ 5 $ edges): the oriented edges sharing a given vertex $ \nu $ either have all their starting point equal $ \nu $, or have all their terminal point equal $ \nu $

  • [1] Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis And Applications, Lecture Notes in Mathematics, 2074, Springer, Heidelberg, 2013, 1–47. doi: 10.1007/978-3-642-36433-4_1.
    [2] Y. AchdouF. CamilliA. Cutrì and N. Tchou, Hamilton–Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445.  doi: 10.1007/s00030-012-0158-1.
    [3] Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.  doi: 10.1137/090758477.
    [4] Y. AchdouS. Oudet and N. Tchou, Hamilton-Jacobi equations for optimal control on junctions and networks, ESAIM Control Optim. Calc. Var., 21 (2015), 876-899.  doi: 10.1051/cocv/2014054.
    [5] Y. Achdou and A. Porretta, Convergence of a finite difference scheme to weak solutions of the system of partial differential equation arising in mean field games, SIAM J. Numer. Anal., 54 (2016), 161-186.  doi: 10.1137/15M1015455.
    [6] L. Boccardo, F. Murat and J.-P. Puel, Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique, in Nonlinear Partial Differential Equations and Their Applications, Res. Notes in Math, 84, Pitman, Boston, Mass.-London, 1983, 19–73.
    [7] F. Camilli and C. Marchi, Stationary mean field games systems defined on networks, SIAM J. Control Optim., 54 (2016), 1085-1103.  doi: 10.1137/15M1022082.
    [8] F. CamilliC. Marchi and D. Schieborn, The vanishing viscosity limit for Hamilton-Jacobi equations on networks, J. Differential Equations, 254 (2013), 4122-4143.  doi: 10.1016/j.jde.2013.02.013.
    [9] P. Cardaliaguet, Notes on mean field games, Preprint, 2011.
    [10] R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I-II, Springer, 2017.
    [11] M.-K. Dao, Ph.D. thesis, 2018.
    [12] K.-J. EngelM. Kramar FijavžR. Nagel and E. Sikolya, Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722.  doi: 10.3934/nhm.2008.3.709.
    [13] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.
    [14] N. ForcadelW. Salazar and M. Zaydan, Homogenization of second order discrete model with local perturbation and application to traffic flow, Discrete Contin. Dyn. Syst., 37 (2017), 1437-1487.  doi: 10.3934/dcds.2017060.
    [15] M. I. Freidlin and S.-J. Sheu, Diffusion processes on graphs: Stochastic differential equations, large deviation principle, Probab. Theory Related Fields, 116 (2000), 181-220.  doi: 10.1007/PL00008726.
    [16] M. I. Freidlin and A. D. Wentzell, Diffusion processes on graphs and the averaging principle, Ann. Probab., 21 (1993), 2215-2245.  doi: 10.1214/aop/1176989018.
    [17] M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences, Springfield, MO, 2006.
    [18] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
    [19] D. A. GomesS. Patrizi and V. Voskanyan, On the existence of classical solutions for stationary extended mean field games, Nonlinear Anal., 99 (2014), 49-79.  doi: 10.1016/j.na.2013.12.016.
    [20] D. A. Gomes and E. Pimentel, Local regularity for mean-field games in the whole space, Minimax Theory Appl., 1 (2016), 65-82. 
    [21] D. A. GomesE. A. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the subquadratic case, Comm. Partial Differential Equations, 40 (2015), 40-76.  doi: 10.1080/03605302.2014.903574.
    [22] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Mathematics, 2003, Springer, Berlin, 2011,205–266. doi: 10.1007/978-3-642-14660-2_3.
    [23] M. HuangP. E. Caines and R. P. Malhamé, An invariance principle in large population stochastic dynamic games, J. Syst. Sci. Complex., 20 (2007), 162-172.  doi: 10.1007/s11424-007-9015-4.
    [24] M. HuangP. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ϵ-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.
    [25] M. HuangR. P. Malhamé and P. 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.  doi: 10.4310/CIS.2006.v6.n3.a5.
    [26] C. ImbertR. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166.  doi: 10.1051/cocv/2012002.
    [27] C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér., 50 (2017), 357–448. doi: 10.24033/asens.2323.
    [28] J.-M. Lasry and P.-L. 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.
    [29] 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), 679-684.  doi: 10.1016/j.crma.2006.09.018.
    [30] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.
    [31] P.-L. Lions and P. Souganidis, Viscosity solutions for junctions: Well posedness and stability, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), 535-545.  doi: 10.4171/RLM/747.
    [32] P.-L. Lions and P. Souganidis, Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 807-816.  doi: 10.4171/RLM/786.
    [33] A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Arch. Rational Mech. Anal., 216 (2015), 1-62.  doi: 10.1007/s00205-014-0799-9.
    [34] A. Porretta, On the weak theory for mean field games systems, Boll. U.M.I., 10 (2017), 411-439.  doi: 10.1007/s40574-016-0105-x.
    [35] J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.  doi: 10.1016/0022-0396(88)90158-1.
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