American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations
  • NHM Home
  • This Issue
  • Next Article
    Compressible and viscous two-phase flow in porous media based on mixture theory formulation
September  2019, 14(3): 537-566. doi: 10.3934/nhm.2019021

A class of infinite horizon mean field games on networks

Yves Achdou 1,, , Manh-Khang Dao 2, , Olivier Ley 3, and  Nicoletta Tchou 2,

1. 

Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris, France
2. 

Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
3. 

Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

* Corresponding author: Yves Achdou

Received  July 2018 Revised  January 2019 Published  May 2019

Fund Project: The authors were partially supported by ANR project ANR-16-CE40-0015-01. The work of O. Ley and N. Tchou was partially supported by the Centre Henri Lebesgue ANR-11-LABX-0020-01.

We consider stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the invariant measure $ m $, a value function $ u $, and the ergodic constant $ \rho $. The function $ u $ is continuous and satisfies general Kirchhoff conditions at the vertices. The invariant measure $ m $ satisfies dual transmission conditions: in particular, $ m $ is discontinuous across the vertices in general, and the values of $ m $ on each side of the vertices satisfy special compatibility conditions. Existence and uniqueness are proven under suitable assumptions.

Keywords: Mean field games, networks, ergodic problems, Kirchhoff conditions.
Mathematics Subject Classification: Primary: 35R02, 49N70; Secondary: 91A13.
Citation: Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks & Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021
References:
[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. doi: 10.1007/978-3-642-36433-4_1.  Google Scholar

[2]

Y. Achdou, F. Camilli, A. Cutrì and N. Tchou, Hamilton–Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445.  doi: 10.1007/s00030-012-0158-1.  Google Scholar

[3]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.  doi: 10.1137/090758477.  Google Scholar

[4]

Y. Achdou, S. Oudet and N. Tchou, Hamilton-Jacobi equations for optimal control on junctions and networks, ESAIM Control Optim. Calc. Var., 21 (2015), 876-899.  doi: 10.1051/cocv/2014054.  Google Scholar

[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.  doi: 10.1137/15M1015455.  Google Scholar

[6]

L. Boccardo, F. Murat and J.-P. Puel, Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique, in Nonlinear Partial Differential Equations and Their Applications, Res. Notes in Math, 84, Pitman, Boston, Mass.-London, 1983, 19–73.  Google Scholar

[7]

F. Camilli and C. Marchi, Stationary mean field games systems defined on networks, SIAM J. Control Optim., 54 (2016), 1085-1103.  doi: 10.1137/15M1022082.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.02.013.  Google Scholar

[9]

P. Cardaliaguet, Notes on mean field games, Preprint, 2011. Google Scholar

[10]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I-II, Springer, 2017.  Google Scholar

[11]

M.-K. Dao, Ph.D. thesis, 2018. Google Scholar

[12]

K.-J. Engel, M. Kramar Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722.  doi: 10.3934/nhm.2008.3.709.  Google Scholar

[13]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.  Google Scholar

[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.  doi: 10.3934/dcds.2017060.  Google Scholar

[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.  doi: 10.1007/PL00008726.  Google Scholar

[16]

M. I. Freidlin and A. D. Wentzell, Diffusion processes on graphs and the averaging principle, Ann. Probab., 21 (1993), 2215-2245.  doi: 10.1214/aop/1176989018.  Google Scholar

[17]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences, Springfield, MO, 2006.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1016/j.na.2013.12.016.  Google Scholar

[20]

D. A. Gomes and E. Pimentel, Local regularity for mean-field games in the whole space, Minimax Theory Appl., 1 (2016), 65-82.   Google Scholar

[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.  doi: 10.1080/03605302.2014.903574.  Google Scholar

[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. doi: 10.1007/978-3-642-14660-2_3.  Google Scholar

[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.  doi: 10.1007/s11424-007-9015-4.  Google Scholar

[24]

M. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ϵ-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar

[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.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[26]

C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166.  doi: 10.1051/cocv/2012002.  Google Scholar

[27]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér., 50 (2017), 357–448. doi: 10.24033/asens.2323.  Google Scholar

[28]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. Ⅰ. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[29]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. Ⅱ. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[30]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar

[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.  doi: 10.4171/RLM/747.  Google Scholar

[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.  doi: 10.4171/RLM/786.  Google Scholar

[33]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Arch. Rational Mech. Anal., 216 (2015), 1-62.  doi: 10.1007/s00205-014-0799-9.  Google Scholar

[34]

A. Porretta, On the weak theory for mean field games systems, Boll. U.M.I., 10 (2017), 411-439.  doi: 10.1007/s40574-016-0105-x.  Google Scholar

[35]

J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.  doi: 10.1016/0022-0396(88)90158-1.  Google Scholar

show all references

References:
[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. doi: 10.1007/978-3-642-36433-4_1.  Google Scholar

[2]

Y. Achdou, F. Camilli, A. Cutrì and N. Tchou, Hamilton–Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445.  doi: 10.1007/s00030-012-0158-1.  Google Scholar

[3]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.  doi: 10.1137/090758477.  Google Scholar

[4]

Y. Achdou, S. Oudet and N. Tchou, Hamilton-Jacobi equations for optimal control on junctions and networks, ESAIM Control Optim. Calc. Var., 21 (2015), 876-899.  doi: 10.1051/cocv/2014054.  Google Scholar

[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.  doi: 10.1137/15M1015455.  Google Scholar

[6]

L. Boccardo, F. Murat and J.-P. Puel, Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique, in Nonlinear Partial Differential Equations and Their Applications, Res. Notes in Math, 84, Pitman, Boston, Mass.-London, 1983, 19–73.  Google Scholar

[7]

F. Camilli and C. Marchi, Stationary mean field games systems defined on networks, SIAM J. Control Optim., 54 (2016), 1085-1103.  doi: 10.1137/15M1022082.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.02.013.  Google Scholar

[9]

P. Cardaliaguet, Notes on mean field games, Preprint, 2011. Google Scholar

[10]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I-II, Springer, 2017.  Google Scholar

[11]

M.-K. Dao, Ph.D. thesis, 2018. Google Scholar

[12]

K.-J. Engel, M. Kramar Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722.  doi: 10.3934/nhm.2008.3.709.  Google Scholar

[13]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.  Google Scholar

[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.  doi: 10.3934/dcds.2017060.  Google Scholar

[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.  doi: 10.1007/PL00008726.  Google Scholar

[16]

M. I. Freidlin and A. D. Wentzell, Diffusion processes on graphs and the averaging principle, Ann. Probab., 21 (1993), 2215-2245.  doi: 10.1214/aop/1176989018.  Google Scholar

[17]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences, Springfield, MO, 2006.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1016/j.na.2013.12.016.  Google Scholar

[20]

D. A. Gomes and E. Pimentel, Local regularity for mean-field games in the whole space, Minimax Theory Appl., 1 (2016), 65-82.   Google Scholar

[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.  doi: 10.1080/03605302.2014.903574.  Google Scholar

[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. doi: 10.1007/978-3-642-14660-2_3.  Google Scholar

[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.  doi: 10.1007/s11424-007-9015-4.  Google Scholar

[24]

M. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ϵ-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar

[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.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[26]

C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166.  doi: 10.1051/cocv/2012002.  Google Scholar

[27]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér., 50 (2017), 357–448. doi: 10.24033/asens.2323.  Google Scholar

[28]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. Ⅰ. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[29]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. Ⅱ. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[30]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar

[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.  doi: 10.4171/RLM/747.  Google Scholar

[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.  doi: 10.4171/RLM/786.  Google Scholar

[33]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Arch. Rational Mech. Anal., 216 (2015), 1-62.  doi: 10.1007/s00205-014-0799-9.  Google Scholar

[34]

A. Porretta, On the weak theory for mean field games systems, Boll. U.M.I., 10 (2017), 411-439.  doi: 10.1007/s40574-016-0105-x.  Google Scholar

[35]

J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.  doi: 10.1016/0022-0396(88)90158-1.  Google Scholar

Figure 1.  Left: the network $ \Gamma $ in which the edges are oriented toward the vertex with larger index ($ 4 $ vertices and $ 4 $ edges). Right: a new network $ \tilde \Gamma $ obtained by adding an artificial vertex ($ 5 $ vertices and $ 5 $ edges): the oriented edges sharing a given vertex $ \nu $ either have all their starting point equal $ \nu $, or have all their terminal point equal $ \nu $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173
[2]

René Carmona, Kenza Hamidouche, Mathieu Laurière, Zongjun Tan. Linear-quadratic zero-sum mean-field type games: Optimality conditions and policy optimization. Journal of Dynamics & Games, 2021, 8 (4) : 403-443. doi: 10.3934/jdg.2021023
[3]

Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279
[4]

Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics & Games, 2019, 6 (3) : 221-239. doi: 10.3934/jdg.2019016
[5]

Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311
[6]

Adriano Festa, Diogo Gomes, Francisco J. Silva, Daniela Tonon. Preface: Mean field games: New trends and applications. Journal of Dynamics & Games, 2021, 8 (4) : i-ii. doi: 10.3934/jdg.2021025
[7]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021, 8 (4) : 445-465. doi: 10.3934/jdg.2021006
[8]

Lucio Boccardo, Luigi Orsina. The duality method for mean field games systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022021
[9]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243
[10]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303
[11]

Yves Achdou, Victor Perez. Iterative strategies for solving linearized discrete mean field games systems. Networks & Heterogeneous Media, 2012, 7 (2) : 197-217. doi: 10.3934/nhm.2012.7.197
[12]

Matt Barker. From mean field games to the best reply strategy in a stochastic framework. Journal of Dynamics & Games, 2019, 6 (4) : 291-314. doi: 10.3934/jdg.2019020
[13]

Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks & Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315
[14]

Juan Pablo Maldonado López. Discrete time mean field games: The short-stage limit. Journal of Dynamics & Games, 2015, 2 (1) : 89-101. doi: 10.3934/jdg.2015.2.89
[15]

Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2021, 8 (1) : 35-59. doi: 10.3934/jdg.2020033
[16]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics & Games, 2021, 8 (4) : 467-486. doi: 10.3934/jdg.2021014
[17]

Kuang Huang, Xuan Di, Qiang Du, Xi Chen. A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4869-4903. doi: 10.3934/dcdsb.2020131
[18]

Michael Herty, Lorenzo Pareschi, Giuseppe Visconti. Mean field models for large data–clustering problems. Networks & Heterogeneous Media, 2020, 15 (3) : 463-487. doi: 10.3934/nhm.2020027
[19]

Max-Olivier Hongler. Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments. Journal of Dynamics & Games, 2020, 7 (1) : 1-20. doi: 10.3934/jdg.2020001
[20]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (248)
  • HTML views (278)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy