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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 and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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 $
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