The planning problem in mean field games as regularized mass transport

Calculus of Variations and Partial Differential Equations

Volumn 58, Issue 3, 2019, Pages

  Graber, P Jameson ^a   Mészáros, Alpár R ^b   Silva, Francisco J ^c   Tonon, Daniela ^d  

^a BAYLOR UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITY OF LIMOGES   (France)

^d UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85067201545     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-019-1561-9     Document Type: Article

References (28)

1. Achdou, Y., Camilli, F., Capuzzo-Dolcetta, I.: Mean field games: numerical methods for the planning problem. SIAM J. Control Optim. 50(1), 77–109 (2012)
2. Ambrosio, L., Figalli, A.: On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations. Calc. Var. Partial Differ. Equ. 31(4), 497–509 (2008)
3. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, New York (2000)
4. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)
5. Brenier, Y.: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Commun. Pure Appl. Math. 52(4), 411–452 (1999)
6. Cardaliaguet, P.: Weak solutions for first order mean field games with local coupling. In: Bettiol, P., Cannarsa, P., Colombo, G., Motta, M., Rampazzo, F. (eds.) Analysis and Geometry in Control Theory and Its Applications, volume 11 of Springer INdAM Series, pp. 111–158. Springer, Cham (2015)
7. Cardaliaguet, P., Carlier, G., Nazaret, B.: Geodesics for a class of distances in the space of probability measures. Calc. Var. Partial Differ. Equ. 48(3–4), 395–420 (2013)
8. Cardaliaguet, P., Graber, P.J.: Mean field games systems of first order. ESAIM Control Optim. Calc. Var. 21(3), 690–722 (2015)
9. Cardaliaguet, P., Graber, P.J., Porretta, A., Tonon, D.: Second order mean field games with degenerate diffusion and local coupling. Nonlinear Differ. Equ. Appl. 22(5), 1287–1317 (2015)
10. Cardaliaguet, P., Mészáros, A.R., Santambrogio, F.: First order mean field games with density constraints: pressure equals price. SIAM J. Control Optim. 54(5), 2672–2709 (2016)
11. Cardaliaguet, P., Silvestre, L.: Hölder continuity to Hamilton–Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side. Commun. PDE 37(9), 1668–1688 (2012)
12. Dolbeault, J., Nazaret, B., Savaré, G.: A new class of transport distances between measures. Calc. Var. Partial Differ. Equ. 34, 193–231 (2009)
13. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Translated from the French. Studies in Mathematics and Its Applications, vol. 1. North-Holland Publishing Co., Amsterdam (1976)
14. Graber, P.J., Mészáros, A.R.: Sobolev regularity for first order mean field games. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(6), 1557–1576 (2018)
15. Gomes, D.A., Seneci, T.: Displacement convexity for first-order mean-field games. Minim. Theory Appl. 3(2), 261–284 (2018)
16. Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006)
17. Lions, P.-L.: Cours au Collège de France. www.college-de-france.fr (Lectures on Nov 27th, Dec 4th–11th, 2009)
18. Liu, J., Loeper, G.: Optimal transport with discrete long range mean field interactions. arXiv:1809.07432
19. Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343, 619–625 (2006)
20. Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343, 679–684 (2006)
21. Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007)
22. ∞ bounds via flow interchange techniques and applications to variational mean field games. Commun. PDE 43(12), 1761–1802 (2018)
23. Orrieri, C., Porretta, A., Savaré, G.: A variational approach to the mean field planning problem. J. Funct. Anal. (2019). 10.1016/j.jfa.2019.04.011
24. Porretta, A.: On the planning problem for a class of mean field games. C. R. Math. Acad. Sci. Paris 351(11–12), 457–462 (2013)
25. Porretta, A.: On the planning problem for the mean field games system. Dyn. Games Appl. 4(2), 231–256 (2014)
26. Prosinski, A., Santambrogio, F.: Global-in-time regularity via duality for congestion-penalized mean field games. Stochastics 89(6–7), 923–942 (2017)
27. Rockafellar, R.T.: Integrals which are convex functionals. II. Pac. J. Math. 39(2), 439–469 (1971)
28. Santambrogio, F.: Regularity via duality in calculus of variations and degenerate elliptic PDEs. J. Math. Anal. Appl. 457(2), 1649–1674 (2018)