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Introduction to Variational Methods for Viscous Ergodic Mean-Field Games with Local Coupling

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Contemporary Research in Elliptic PDEs and Related Topics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 33))

Abstract

We collect in these notes some results on the existence and uniqueness of classical solutions to viscous ergodic Mean-Field Game systems with local coupling. We present in particular some methods and ideas based on convex optimization techniques and elliptic regularity.

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Notes

  1. 1.

    This solution m can be found for example by standard methods in calculus of variations, i.e. by minimizing the convex functional \(m \mapsto \frac {1}{2} \int _Q |\nabla m|{ }^2 - ({\mathrm {div}} w)m\) subject to the constraint ∫Q m = 1; regularity of the minimizing weak solution is classical by uniform ellipticity.

  2. 2.

    Note that L(⋅) here coincides with L as in the introduction. Indeed, \(\tilde L = H^* = (L^*)^* = L\).

References

  1. S. Agmon, The L p approach to the Dirichlet problem. I. Regularity theorems. Ann. Scuola Norm. Sup. Pisa (3) 13, 405–448 (1959)

    Google Scholar 

  2. M. Bardi, E. Feleqi, The derivation of ergodic mean field game equations for several populations of players. Dyn. Games Appl. 3(4), 523–536 (2013)

    Article  MathSciNet  Google Scholar 

  3. M. Bardi, E. Feleqi, Nonlinear elliptic systems and mean field games. NoDEA Nonlinear Differ. Equ. Appl. 23, 23–44 (2016)

    Article  MathSciNet  Google Scholar 

  4. G. Barles, P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM J. Math. Anal. 32(6), 1311–1323 (2001)

    Article  MathSciNet  Google Scholar 

  5. J.-D. Benamou, G. Carlier, F. Santambrogio, Variational Mean Field Games, ed. by Bellomo, Degond, Tadmor. Active Particles, vol. 1 (Springer, Berlin, 2017)

    Google Scholar 

  6. A. Bensoussan, Perturbation Methods in Optimal Control (John Wiley & Sons, Hoboken, 1988)

    MATH  Google Scholar 

  7. A. Bensoussan, J. Y. Frehse, Mean Field Games and Mean Field Type Control (Springer, Berlin, 2013)

    Book  Google Scholar 

  8. J. Borwein, J. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples (Cambridge University Press, Cambridge, 2010)

    Book  Google Scholar 

  9. A. Briani, P. Cardaliaguet, Stable solutions in potential mean field game systems. Nonlinear Differ. Equ. Appl. 25, 1 (2018)

    Article  MathSciNet  Google Scholar 

  10. P. Cardaliaguet, Notes on mean-field games (2011). Available at https://www.ceremade.dauphine.fr/ cardaliaguet/MFG20130420.pdf

    Google Scholar 

  11. P. Cardaliaguet, P.-J. Graber, Mean field games systems of first order. ESAIM Control Optim. Calc. Var. 21, 690–722 (2015)

    Article  MathSciNet  Google Scholar 

  12. P. Cardaliaguet, P.J. Graber, A. Porretta, D. Tonon, Second order mean field games with degenerate diffusion and local coupling. NoDEA Nonlinear Differ. Equ. Appl. 22(5), 1287–1317 (2015)

    Article  MathSciNet  Google Scholar 

  13. P. Cardaliaguet, A.R. Mészáros, F. Santambrogio, First order mean field games with density constraints: pressure equals price. SIAM J. Control. Optim. 54(5), 2672–2709 (2016)

    Article  MathSciNet  Google Scholar 

  14. P. Cardaliaguet, A. Porretta, D. Tonon, A segregation problem in multi-population mean field games, in Advances in Dynamic and Mean Field Games. ISDG 2016. Annals of the International Society of Dynamic Games ed. by J. Apaloo, B. Viscolani, vol. 15 (Birkhäuser, Basel, 2017), pp. 49–70

    Chapter  Google Scholar 

  15. P. Cardaliaguet, F. Delarue, J.-M. Lasry, P.-L. Lions, The master equation and the convergence problem in mean field games. Preprint arXiv:1509.02505

    Google Scholar 

  16. A. Cesaroni, M. Cirant, Concentration of ground states in stationary mean-field games systems. Anal. PDE 12(3), 737–787 (2019)

    Article  MathSciNet  Google Scholar 

  17. A. Cesaroni, M. Cirant, S. Dipierro, M. Novaga, E. Valdinoci, On stationary fractional mean field games. J. Math. Pures Appl. 122, 1–22 (2019)

    Article  MathSciNet  Google Scholar 

  18. M. Cirant, On the solvability of some ergodic control problems in \({\mathbb R}^d\). SIAM J. Control Optim. 52(6), 4001–4026 (2014)

    Google Scholar 

  19. M. Cirant, Multi-population mean field games systems with Neumann boundary conditions. J. Math. Pures Appl. (9) 103(5), 1294–1315 (2015)

    Article  MathSciNet  Google Scholar 

  20. M. Cirant, A generalization of the Hopf-Cole transformation for stationary mean field games systems. C.R. Math. 353(9), 807–811 (2015)

    Article  MathSciNet  Google Scholar 

  21. M. Cirant, Stationary focusing mean-field games. Commun. Part. Diff. Eq. 41(8), 1324–1346 (2016)

    Article  MathSciNet  Google Scholar 

  22. M. Cirant, D. Tonon, Time-dependent focusing mean-field games: the sub-critical case. J. Dyn. Diff. Equat. 31(1), 49–79 (2019)

    Article  MathSciNet  Google Scholar 

  23. M. Cirant, G. Verzini, Bifurcation and segregation in quadratic two-populations mean field games systems. ESAIM Control Optim. Calc. Var. 23, 1145–1177 (2017)

    Article  MathSciNet  Google Scholar 

  24. I. Ekeland, R. Temam, Convex Analysis and Variational Problems (North-Holland Publishing Co., Amsterdam-Oxford, 1976)

    MATH  Google Scholar 

  25. E. Feleqi, The derivation of ergodic mean field game equations for several populations of players. Dyn. Games Appl. 3(4), 523–536 (2013)

    Article  MathSciNet  Google Scholar 

  26. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. (Springer-Verlag, Berlin, 2001)

    MATH  Google Scholar 

  27. D.A. Gomes, J. Saude, Mean field games models–a brief survey. Dyn. Games Appl. 4(2), 110–154 (2014)

    Article  MathSciNet  Google Scholar 

  28. D.A. Gomes, S. Patrizi, V. Voskanyan, On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. 99, 49–79 (2014)

    Article  MathSciNet  Google Scholar 

  29. D.A. Gomes, L. Nurbekyan, M. Prazeres, One-dimensional stationary mean-field games with local coupling. Dyn. Games Appl. 8(2), 315–351 (2018)

    Article  MathSciNet  Google Scholar 

  30. D.A. Gomes, E.A. Pimentel, V. Voskanyan, Regularity Theory for Mean-Field Game Systems (Springer, Berlin, 2016)

    Book  Google Scholar 

  31. M. Huang, R. Malhamé, P. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006)

    MathSciNet  MATH  Google Scholar 

  32. R. Khasminskii, Stochastic stability of differential equations, in Stochastic Modelling and Applied Probability, vol. 66 (Springer, Berlin, 2012)

    Book  Google Scholar 

  33. J.-M. Lasry, P.-L. Lions, Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. Math. Ann. 283(4), 583–630 (1989)

    Article  MathSciNet  Google Scholar 

  34. J.-M. Lasry, P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)

    Article  Google Scholar 

  35. J.-M. Lasry, P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)

    Article  Google Scholar 

  36. J.-M. Lasry, P.-L. Lions, Mean field games. Japan. J. Math. 2, 229–260 (2007)

    Article  MathSciNet  Google Scholar 

  37. P.-L. Lions, Cours au Collége de France, www.college-de-france.fr

  38. A.R. Mészáros, F.J. Silva, A variational approach to second order mean field games with density constraints: the stationary case. J. Math. Pures Appl. (9) 104(6), 1135–1159 (2015)

    Article  MathSciNet  Google Scholar 

  39. E. Pimentel, V. Voskanyan, Regularity for second-order stationary mean-field games. Indiana Univ. Math. J. 66, 1–22 (2017)

    Article  MathSciNet  Google Scholar 

  40. R.T. Rockafellar, Integral functionals, normal integrands and measurable selections. Lect. Notes Math. 543, 157–207 (1976)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work has been supported by the INdAM Intensive Period “Contemporary Research in elliptic PDEs and related topics”. The authors are partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games”.

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Authors and Affiliations

  1. Dipartimento di Scienze Statistiche, Università di Padova, Padova, Italy

    Annalisa Cesaroni

  2. Dipartimento di Matematica Tullio Levi Civita, Università di Padova, Padova, Italy

    Marco Cirant

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  1. Annalisa Cesaroni
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  2. Marco Cirant
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Correspondence to Annalisa Cesaroni .

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Editors and Affiliations

  1. Dept. of Mathematics and Statistics, University of Western Australia, Crawley, Perth, WA, Australia

    Serena Dipierro

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Cesaroni, A., Cirant, M. (2019). Introduction to Variational Methods for Viscous Ergodic Mean-Field Games with Local Coupling. In: Dipierro, S. (eds) Contemporary Research in Elliptic PDEs and Related Topics. Springer INdAM Series, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-030-18921-1_5

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