American Institute of Mathematical Sciences

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September  2015, 35(9): 4173-4192. doi: 10.3934/dcds.2015.35.4173

A model problem for Mean Field Games on networks

Fabio Camilli 1, , Elisabetta Carlini 2, and  Claudio Marchi 3,

1. 

"Sapienza" Università di Roma, Dip. di Scienze di Base e Applicate per l'Ingegneria, via Scarpa 16, 0161 Roma
2. 

"Sapienza" Università di Roma, Dipartimento di Matematica, P.le A. Moro 5, 00185 Roma, Italy
3. 

Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, 35121 Padova

Received  March 2014 Revised  September 2014 Published  April 2015

In [14], Guéant, Lasry and Lions considered the model problem ``What time does meeting start?'' as a prototype for a general class of optimization problems with a continuum of players, called Mean Field Games problems. In this paper we consider a similar model, but with the dynamics of the agents defined on a network. We discuss appropriate transition conditions at the vertices which give a well posed problem and we present some numerical results.
Keywords: Mean Field Games, numerical methods., stochastic optimal control, Networks.
Mathematics Subject Classification: Primary: 91A15, 35R02; Secondary: 35B30, 49N70, 65M0.
Citation: Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173
References:
[1]

Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math. volume 2074, Springer, Berlin, (2013), 1-47. doi: 10.1007/978-3-642-36433-4_1.

[2]

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.

[3]

J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion, Comm. Partial Differential Equations, 21 (1996), 255-279. doi: 10.1080/03605309608821184.

[4]

M. Burger, M. Di Francesco, P. Markowich and M.-T. Wolfram, Mean field games with linear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333, arXiv:1304.5201. doi: 10.3934/dcdsb.2014.19.1311.

[5]

F. Camilli, C. Marchi and D. Schieborn, The vanishing viscosity limit for Hamilton-Jacobi equation on networks, J. Differential Equations, 254 (2013), 4122-4143. doi: 10.1016/j.jde.2013.02.013.

[6]

P. Cardaliaguet, Notes on Mean Field Games: from P.-L. Lions' lectures at Collège de France, Lecture Notes given at Tor Vergata, 2010.

[7]

G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783. doi: 10.1137/090771417.

[8]

C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520. doi: 10.1016/j.mcm.2010.06.012.

[9]

M. 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.

[10]

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

[11]

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

[12]

D. Gomes and J. Saude, Mean field games - A brief survey, Dyn.Games Appl., 4 (2014), 110-154. doi: 10.1007/s13235-013-0099-2.

[13]

M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240. doi: 10.1007/s00245-006-0887-9.

[14]

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 Math. volume 2003, Springer, Berlin, (2011), 205-266. doi: 10.1007/978-3-642-14660-2_3.

[15]

O. Guéant, Mean field games equations with quadratic Hamiltonian: A specific approach, M3AS Math. Models Methods Appl. Sci., 22 (2012), 1250022, 37pp. doi: 10.1142/S0218202512500224.

[16]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968.

[17]

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

[18]

D. Mugnolo, Gaussian estimates for a heat equation on a network, Netw. Het. Media, 2 (2007), 55-79. doi: 10.3934/nhm.2007.2.55.

[19]

D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations, Math. Methods Appl. Sci., 30 (2007), 681-706. doi: 10.1002/mma.805.

[20]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.

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 Math. volume 2074, Springer, Berlin, (2013), 1-47. doi: 10.1007/978-3-642-36433-4_1.

[2]

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.

[3]

J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion, Comm. Partial Differential Equations, 21 (1996), 255-279. doi: 10.1080/03605309608821184.

[4]

M. Burger, M. Di Francesco, P. Markowich and M.-T. Wolfram, Mean field games with linear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333, arXiv:1304.5201. doi: 10.3934/dcdsb.2014.19.1311.

[5]

F. Camilli, C. Marchi and D. Schieborn, The vanishing viscosity limit for Hamilton-Jacobi equation on networks, J. Differential Equations, 254 (2013), 4122-4143. doi: 10.1016/j.jde.2013.02.013.

[6]

P. Cardaliaguet, Notes on Mean Field Games: from P.-L. Lions' lectures at Collège de France, Lecture Notes given at Tor Vergata, 2010.

[7]

G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783. doi: 10.1137/090771417.

[8]

C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520. doi: 10.1016/j.mcm.2010.06.012.

[9]

M. 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.

[10]

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

[11]

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

[12]

D. Gomes and J. Saude, Mean field games - A brief survey, Dyn.Games Appl., 4 (2014), 110-154. doi: 10.1007/s13235-013-0099-2.

[13]

M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240. doi: 10.1007/s00245-006-0887-9.

[14]

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 Math. volume 2003, Springer, Berlin, (2011), 205-266. doi: 10.1007/978-3-642-14660-2_3.

[15]

O. Guéant, Mean field games equations with quadratic Hamiltonian: A specific approach, M3AS Math. Models Methods Appl. Sci., 22 (2012), 1250022, 37pp. doi: 10.1142/S0218202512500224.

[16]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968.

[17]

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

[18]

D. Mugnolo, Gaussian estimates for a heat equation on a network, Netw. Het. Media, 2 (2007), 55-79. doi: 10.3934/nhm.2007.2.55.

[19]

D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations, Math. Methods Appl. Sci., 30 (2007), 681-706. doi: 10.1002/mma.805.

[20]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.

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