# American Institute of Mathematical Sciences

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

## A model problem for Mean Field Games on networks

 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.
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.

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##### 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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