Well-posedness of mean field games master equations involving non-separable local Hamiltonians

Ambrose, David; Mészáros, Alpár

doi:10.1090/tran/8760

Well-posedness of mean field games master equations involving non-separable local Hamiltonians
HTML articles powered by AMS MathViewer

by David M. Ambrose and Alpár R. Mészáros HTML | PDF

Trans. Amer. Math. Soc. 376 (2023), 2481-2523 Request permission

Abstract:

In this paper we construct short time classical solutions to a class of master equations in the presence of non-degenerate individual noise arising in the theory of mean field games. The considered Hamiltonians are non-separable and local functions of the measure variable, therefore the equation is restricted to absolutely continuous measures whose densities lie in suitable Sobolev spaces. Our results hold for smooth enough Hamiltonians, without any additional structural conditions as convexity or monotonicity.

References

Yves Achdou, Francisco J. Buera, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll, Partial differential equation models in macroeconomics, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 2028, 20130397, 19. MR 3268061, DOI 10.1098/rsta.2013.0397
Yves Achdou and Alessio Porretta, Mean field games with congestion, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 2, 443–480. MR 3765549, DOI 10.1016/j.anihpc.2017.06.001
David M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal. 35 (2003), no. 1, 211–244. MR 2001473, DOI 10.1137/S0036141002403869
David M. Ambrose, Existence theory for non-separable mean field games in Sobolev spaces, Indiana Univ. Math. J. 71 (2022), no. 2, 611–647. MR 4420100, DOI 10.1512/iumj.2022.71.8900
David M. Ambrose, Strong solutions for time-dependent mean field games with non-separable Hamiltonians, J. Math. Pures Appl. (9) 113 (2018), 141–154. MR 3784807, DOI 10.1016/j.matpur.2018.03.003
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. MR 2401600
J. Thomas Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure Appl. Math. 34 (1981), no. 3, 359–392. MR 611750, DOI 10.1002/cpa.3160340305
A. Bensoussan, P. J. Graber, and S. C. P. Yam, Control on Hilbert spaces and application to mean field type control theory, arXiv:2005.10770, 2020.
P. Cardaliaguet, The convergence problem in mean field games with local coupling, Appl. Math. Optim. 76 (2017), no. 1, 177–215. MR 3679342, DOI 10.1007/s00245-017-9434-0
P. Cardaliaguet, M. Cirant, A. Porretta, Splitting methods and short time existence for the master equations in mean field games, J. Eur. Math. Soc. (JEMS), to appear, arXiv:2001.11970, 2020. DOI 10.4171/JEMS/1227.
Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions, The master equation and the convergence problem in mean field games, Annals of Mathematics Studies, vol. 201, Princeton University Press, Princeton, NJ, 2019. MR 3967062, DOI 10.2307/j.ctvckq7qf
René Carmona and François Delarue, Probabilistic theory of mean field games with applications. II, Probability Theory and Stochastic Modelling, vol. 84, Springer, Cham, 2018. Mean field games with common noise and master equations. MR 3753660
J.-F. Chassagneux, D. Crisan, and F. Delarue, A probabilistic approach to classical solutions of the master equation for large population equilibria, Mem. Amer. Math. Soc., to appear.
Marco Cirant, Roberto Gianni, and Paola Mannucci, Short-time existence for a general backward-forward parabolic system arising from mean-field games, Dyn. Games Appl. 10 (2020), no. 1, 100–119. MR 4064664, DOI 10.1007/s13235-019-00311-5
François Delarue, Daniel Lacker, and Kavita Ramanan, From the master equation to mean field game limit theory: a central limit theorem, Electron. J. Probab. 24 (2019), Paper No. 51, 54. MR 3954791, DOI 10.1214/19-EJP298
François Delarue, Daniel Lacker, and Kavita Ramanan, From the master equation to mean field game limit theory: large deviations and concentration of measure, Ann. Probab. 48 (2020), no. 1, 211–263. MR 4079435, DOI 10.1214/19-AOP1359
David Evangelista, Rita Ferreira, Diogo A. Gomes, Levon Nurbekyan, and Vardan Voskanyan, First-order, stationary mean-field games with congestion, Nonlinear Anal. 173 (2018), 37–74. MR 3802565, DOI 10.1016/j.na.2018.03.011
David Evangelista and Diogo A. Gomes, On the existence of solutions for stationary mean-field games with congestion, J. Dynam. Differential Equations 30 (2018), no. 4, 1365–1388. MR 3871606, DOI 10.1007/s10884-017-9615-1
W. Gangbo and A. R. Mészáros, Global well-posedness of Master equations for deterministic displacement convex potential mean field games, Comm. Pure Appl. Math., to appear. arXiv:2004.01660 (2020). DOI 10.1002/cpa.22069.
W. Gangbo, A. R. Mészáros, C. Mou, and J. Zhang, Mean field games master equations with non-separable Hamiltonians and displacement monotonicity, Ann. Probab. 50 (2022), no. 6, 2178–2217.
Wilfrid Gangbo and Adrian Tudorascu, On differentiability in the Wasserstein space and well-posedness for Hamilton-Jacobi equations, J. Math. Pures Appl. (9) 125 (2019), 119–174 (English, with English and French summaries). MR 3944201, DOI 10.1016/j.matpur.2018.09.003
Wilfrid Gangbo and Adrian Tudorascu, Weak KAM theory on the Wasserstein torus with multidimensional underlying space, Comm. Pure Appl. Math. 67 (2014), no. 3, 408–463. MR 3158572, DOI 10.1002/cpa.21492
Wilfrid Gangbo and Andrzej Święch, Existence of a solution to an equation arising from the theory of mean field games, J. Differential Equations 259 (2015), no. 11, 6573–6643. MR 3397332, DOI 10.1016/j.jde.2015.08.001
Diogo Gomes and Marc Sedjro, One-dimensional, forward-forward mean-field games with congestion, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 5, 901–914. MR 3817560, DOI 10.3934/dcdss.2018054
Diogo A. Gomes and Vardan K. Voskanyan, Short-time existence of solutions for mean-field games with congestion, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 778–799. MR 3431662, DOI 10.1112/jlms/jdv052
Minyi Huang, Roland P. Malhamé, and Peter E. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst. 6 (2006), no. 3, 221–251. MR 2346927, DOI 10.4310/CIS.2006.v6.n3.a5
Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris 343 (2006), no. 9, 619–625 (French, with English and French summaries). MR 2269875, DOI 10.1016/j.crma.2006.09.019
Jean-Michel Lasry and Pierre-Louis Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris 343 (2006), no. 10, 679–684 (French, with English and French summaries). MR 2271747, DOI 10.1016/j.crma.2006.09.018
Jean-Michel Lasry and Pierre-Louis Lions, Mean field games, Jpn. J. Math. 2 (2007), no. 1, 229–260. MR 2295621, DOI 10.1007/s11537-007-0657-8
P.-L. Lions, Lectures at Collège de France.
Sergio Mayorga, Short time solution to the master equation of a first order mean field game, J. Differential Equations 268 (2020), no. 10, 6251–6318. MR 4069006, DOI 10.1016/j.jde.2019.11.031
Stéphane Mischler and Clément Mouhot, Kac’s program in kinetic theory, Invent. Math. 193 (2013), no. 1, 1–147. MR 3069113, DOI 10.1007/s00222-012-0422-3
C. Mou and J. Zhang, Wellposedness of second order master equations for mean field games with nonsmooth data, Mem. Amer. Math. Soc., to appear. arXiv:1903.09907v3 (2019).
Rémi Peyre, Comparison between $\rm W_2$ distance and $\dot \textrm {H}^{-1}$ norm, and localization of Wasserstein distance, ESAIM Control Optim. Calc. Var. 24 (2018), no. 4, 1489–1501. MR 3922440, DOI 10.1051/cocv/2017050