A class of infinite horizon mean field games on networks

Networks and Heterogeneous Media

Volumn 14, Issue 3, 2019, Pages 537-566

  Achdou, Yves ^a   Dao, Manh Khang ^b   Ley, Olivier ^b   Tchou, Nicoletta ^b  

^a UNIVERSITÉ PARIS DESCARTES   (France)

^b UNIVERSITÉ DE RENNES 1   (France)

Author keywords

Ergodic problems; Kirchhoff conditions; Mean field games; Networks

Indexed keywords

DYNAMIC PROGRAMMING; NETWORKS (CIRCUITS); STOCHASTIC SYSTEMS;

COMPATIBILITY CONDITIONS; ERGODICS; EXISTENCE AND UNIQUENESS; HAMILTON JACOBI BELLMAN EQUATION; INVARIANT MEASURE; KIRCHHOFF; MEAN FIELD GAMES; TRANSMISSION CONDITIONS;

FOKKER PLANCK EQUATION;

EID: 85068609345     PISSN: 15561801     EISSN: 1556181X     Source Type: Journal    
DOI: 10.3934/nhm.2019021     Document Type: Article

References (35)

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
2. Y. Achdou, F. Camilli, A. Cutri and N. Tchou Hamilton-Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445
3. Y. Achdou and I. Capuzzo-Dolcetta Mean field games: numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162
4. Y. Achdou, S. Oudet and N. Tchou Hamilton-Jacobi equations for optimal control on junc-tions and networks, ESAIM Control Optim. Calc. Var., 21 (2015), 876-899
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
6. L. Boccardo, F. Murat and J.-P. Puel Existence de solutions faibles pour des équations el-liptiques quasi-linéaires a 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
8. F. Camilli, C. Marchi and D. Schieborn The vanishing viscosity limit for Hamilton-Jacobi equations on networks, J. Differential Equations, 254 (2013), 4122-4143
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. Engel, M. Kramar Fijavz, R. Nagel and E. Sikolya Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722
13. A. Ern and J.-L. Guermond Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159, Springer-Verlag, New York, 2004
14. N. Forcadel, W. 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
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
16. M. I. Freidlin and A. D. Wentzell Diffusion processes on graphs and the averaging principle, Ann. Probab., 21 (1993), 2215-2245
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. Gomes, S. Patrizi and V. Voskanyan On the existence of classical solutions for stationary extended mean field games, Nonlinear Anal., 99 (2014), 49-79
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. Gomes, E. A. Pimentel and H. Sánchez-Morgado Time-dependent mean-field games in the subquadratic case, Comm. Partial Differential Equations, 40 (2015), 40-76
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
23. M. Huang, P. E. Caines and R. P. Malhamé An invariance principle in large population stochastic dynamic games, J. Syst. Sci. Complex., 20 (2007), 162-172
24. M. Huang, P. E. Caines and R. P. Malhamé Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ψ-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571
25. M. Huang, R. 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
26. C. Imbert, R. Monneau and H. Zidani A Hamilton-Jacobi approach to junction problems and application to traffic ows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166
27. C. Imbert and R. Monneau Flux-limited solutions for quasi-convex Hamilton-Jacobi equa-tions on networks, Ann. Sci. Ec. Norm. Super., 50 (2017), 357-448
28. J.-M. Lasry and P.-L. Lions Jeuxa champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625
29. J.-M. Lasry and P.-L. Lions Jeuxa champ moyen. II. Horizon fini et contr^ole optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684
30. J.-M. Lasry and P.-L. Lions Mean field games, Jpn. J. Math., 2 (2007), 229-260
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
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
33. A. Porretta Weak solutions to Fokker-Planck equations and mean field games, Arch. Rational Mech. Anal., 216 (2015), 1-62
34. A. Porretta On the weak theory for mean field games systems, Boll. U.M.I., 10 (2017), 411-439
35. J. von Below Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337