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A class of infinite horizon mean field games on networks
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 |
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.
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] | |
[10] |
R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I-II, Springer, 2017. |
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[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] | |
[10] |
R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I-II, Springer, 2017. |
[11] | |
[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. |

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