On the long time convergence of potential MFG

Nonlinear Differential Equations and Applications

Volumn 26, Issue 2, 2019, Pages

  Masoero, Marco ^a  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

Ergodicity; Mean field games; PDE control; Weak KAM theory

Indexed keywords

EID: 85064175938     PISSN: 10219722     EISSN: 14209004     Source Type: Journal    
DOI: 10.1007/s00030-019-0560-z     Document Type: Article

References (37)

1. Barles, G., Souganidis, P.E.: Space–time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM J. Math. Anal. 32(6), 1311–1323 (2001)
2. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numerische Mathematik 84(3), 375–393 (2000)
3. Bogachev, V.I., Krylov, N.V., Röckner, M.: Elliptic and parabolic equations for measures. Russ. Math. Surv. 64(6), 973 (2009)
4. Bogachev, V.I., Krylov, N.V., Röckner, M., Shaposhnikov, S.V.: Fokker–Planck–Kolmogorov Equations, vol. 207. American Mathematical Society, Providence (2015)
5. Briani, A., Cardaliaguet, P.: Stable solutions in potential mean field game systems. Nonlinear Differ. Equ. Appl. NoDEA 25(1), 1 (2018)
6. Cagnetti, F., Gomes, D., Mitake, H., Tran, H.V.: A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians. Annales de l’Institut Henri Poincare (C) Non Linear Analysis. 32(1), 183–200 (2015)
7. Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control, vol. 58. Springer, Berlin (2004)
8. Cardaliaguet, P.: Notes on mean field games. Tech. report, Technical report (2010)
9. Cardaliaguet, P.: Long time average of first order mean field games and weak kam theory. Dyn. Games Appl. 3(4), 473–488 (2013)
10. Cardaliaguet, P., François, D., Lasry, J.-M., Lions, P.-L.: The master equation and the convergence problem in mean field games. arXiv preprint arXiv:1509.02505 (2015)
11. Cardaliaguet, P., Graber, P.J., Porretta, A., Tonon, D.: Second order mean field games with degenerate diffusion and local coupling. Nonlinear Differ. Equ. Appl. NoDEA 22(5), 1287–1317 (2015)
12. Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A.: Long time average of mean field games with a nonlocal coupling. SIAM J. Control Optim. 51(5), 3558–3591 (2013)
13. Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A., et al.: Long time average of mean field games. NHM 7(2), 279–301 (2012)
14. Cardaliaguet, P., Porretta, A.: Long time behavior of the master equation in mean field game theory. Anal. PDE 12(6), 1397–1453 (2019)
15. Cirant, M., Nurbekyan, L.: The variational structure and time-periodic solutions for mean-field games systems. arXiv preprint arXiv:1804.08943 (2018)
16. Cirant, M.: On the existence of oscillating solutions in non-monotone mean-field games. J. Differ. Equ. 266(12), 8067–8093 (2019)
17. De Vries, J.: Elements of Topological Dynamics, vol. 257. Springer, Berlin (2013)
18. Fathi, A.: Solutions kam faibles conjuguées et barrieres de peierls. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 325(6), 649–652 (1997)
19. Fathi, A.: Théoreme kam faible et théorie de mather sur les systemes lagrangiens. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 324(9), 1043–1046 (1997)
20. Fathi, A.: Sur la convergence du semi-groupe de lax-oleinik. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 327(3), 267–270 (1998)
21. Fathi, A.: Weak KAM theorem in Lagrangian dynamics preliminary version number 10, by CUP, Cambridge University Press (2008)
22. Ferreira, R., Gomes, D.A.: On the convergence of finite state mean-field games through γ -convergence. J. Math. Anal. Appl. 418(1), 211–230 (2014)
23. Gangbo, W., Tudorascu, A.: Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem. Adv. Math. 224(1), 260–292 (2010)
24. Gangbo, W., Tudorascu, A.: Weak KAM theory on the Wasserstein torus with multidimensional underlying space. Commun. Pure Appl. Math. 67(3), 408–463 (2014)
25. Gomes, D.A., Mohr, J., Souza, R.R.: Discrete time, finite state space mean field games. Journal de mathématiques pures et appliquées 93(3), 308–328 (2010)
26. Gomes, D., Nurbekyan, L.: On the minimizers of calculus of variations problems in Hilbert spaces. Calc. Var. Partial Differ. Equ. 52(1–2), 65–93 (2015)
27. Gomes, D., Nurbekyan, L.: An infinite-dimensional weak KAM theory via random variables. Discrete Contin. Dyn. Syst. A 36(11), 6167–6185 (2016)
28. Gomes, D., Sedjro, M.: One-dimensional, forward-forward mean-field games with congestion. Discrete Contin. Dyn. Syst. Ser. S 11, 901 (2017)
29. Gomes, D.A., Valdinoci, E.: Duality theory, representation formulas and uniqueness results for viscosity solutions of Hamilton-Jacobi equations. In: Peixoto, M., Pinto, A., Rand, D. (eds.) Dynamics, Games and Science II, pp. 361–386. Springer, Berlin (2011)
30. Guéant, O., Lasry, J.-M., Lions, P.-L.: Mean field games and applications. In: Paris-Princeton Lectures on Mathematical Finance 2010, pp. 205–266. Springer, Berlin (2011)
31. Huang, M., Malhamé, R.P., Caines, P.E., et al.: Large population stochastic dynamic games: closed-loop Mckean–Vlasov systems and the nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–252 (2006)
32. Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasi-linear Equations of Parabolic Type, vol. 23. American Mathematical Society, Providence (1988)
33. Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. i—le cas stationnaire. Comptes Rendus Mathématique 343(9), 619–625 (2006)
34. Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. ii—horizon fini et contrôle optimal. Comptes Rendus Mathématique 343(10), 679–684 (2006)
35. Lions, P.-L.: Mean-field games. cours au collège de france (2007–2008)
36. Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton–Jacobi equations. Unpublished preprint (1987)
37. Mészáros, A.R., Silva, F.J.: On the variational formulation of some stationary second-order mean field games systems. SIAM J. Math. Anal. 50(1), 1255–1277 (2018)