April  2014, 34(4): 1397-1441. doi: 10.3934/dcds.2014.34.1397

Optimal transport and large number of particles

1. 

School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, United States

Received  January 2013 Revised  April 2013 Published  October 2013

We present an approach for proving uniqueness of ODEs in the Wasserstein space. We give an overview of basic tools needed to deal with Hamiltonian ODE in the Wasserstein space and show various continuity results for value functions. We discuss a concept of viscosity solutions of Hamilton-Jacobi equations in metric spaces and in some cases relate it to viscosity solutions in the sense of differentials in the Wasserstein space.
Citation: Wilfrid Gangbo, Andrzej Świech. Optimal transport and large number of particles. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1397-1441. doi: 10.3934/dcds.2014.34.1397
References:
[1]

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

[2]

Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109. doi: 10.1137/100790069.

[3]

G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in $ R^n $ , Math. Z., 230 (1999), 259-316. doi: 10.1007/PL00004691.

[4]

L. Ambrosio and J. Feng, On a class of first order Hamilton-Jacobi equations in metric spaces,, preprint., (). 

[5]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.

[6]

L. Ambrosio and W. Gangbo, Hamiltonian ODEs in the Wasserstein space of probability measures, Comm. Pure Appl. Math., 61 (2007), 18-53. doi: 10.1002/cpa.20188.

[7]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.

[8]

L. Ambrosio, N. Gigli and G. Savaré, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 969-996. doi: 10.4171/RMI/746.

[9]

V. Barbu and G. Da Prato, "Hamilton-Jacobi Equations in Hilbert Spaces," Research Notes in Mathematics, 86, Pitman (Advanced Publishing Program), Boston, MA, 1983.

[10]

P. Bernard and B. Buffoni, Optimal mass transportation and Mather theory, Journal of the European Mathematical Society, 9 (2007), 85-121. doi: 10.4171/JEMS/74.

[11]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Macroscopic fluctuation theory for stationary non-equilibrium states, J. Statist. Phys., 107 (2002), 635-675. doi: 10.1023/A:1014525911391.

[12]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Large deviations for the boundary driven symmetric simple exclusion process, Math. Phys. Anal. Geom., 6 (2003), 231-267. doi: 10.1023/A:1024967818899.

[13]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Minimum dissipation principle in stationary non-equilibrium states, J. Statist. Phys., 116 (2004), 831-841. doi: 10.1023/B:JOSS.0000037220.57358.94.

[14]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Stochastic interacting particle systems out of equilibrium, J. Stat. Mech. Theory Exp., 2007 (2007), P07014, 35 pp.

[15]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Action functional and quasi-potential for the Burgers equation in a bounded interval, Comm. Pure Appl. Math., 64 (2011), 649-696. doi: 10.1002/cpa.20357.

[16]

L. Bertini, D. Gabrielli and J. L. Lebowitz, Large deviations for a stochastic model of heat flow, J. Stat. Phys., 121 (2005), 843-885. doi: 10.1007/s10955-005-5527-2.

[17]

A. J. Bertozzi and A. Majda, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.

[18]

J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc., 303 (1987), 517-527. doi: 10.1090/S0002-9947-1987-0902782-7.

[19]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402.

[20]

Y. Brenier, W. Gangbo, G. Savaré and M. Westdickenberg, Sticky particle dynamics with interactions, Journal de Math. Pures et Appliquées (9), 99 (2013), 577-617. doi: 10.1016/j.matpur.2012.09.013.

[21]

P. Cardialaguet, J.-M. Lasry, P.-L. Lions and A. Porretta, Long times average of mean field games, Networks and Heterogeneous Media, 7 (2012), 279-301. doi: 10.3934/nhm.2012.7.279.

[22]

P. Cardialaguet and M. Quincampoix, Deterministic differential games under probability knowledge of initial condition, Int. Game Theory Rev., 10 (2008), 1-16. doi: 10.1142/S021919890800173X.

[23]

M. G. Crandall, H. Ishii and P.-L. Lions, User's Guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.

[24]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in Banach spaces, in "Trends in the Theory and Practice of Nonlinear Analysis" (Arlington, Tex., 1984), North-Holland Math. Stud., 110, North-Holland, Amsterdam, (1985), 115-119. doi: 10.1016/S0304-0208(08)72698-7.

[25]

M. G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. I. Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), 379-396. doi: 10.1016/0022-1236(85)90011-4.

[26]

M. G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions, J. Funct. Anal., 63 (1986), 368-405. doi: 10.1016/0022-1236(86)90026-1.

[27]

M. G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. III, J. Funct. Anal., 68 (1986), 214-247. doi: 10.1016/0022-1236(86)90005-4.

[28]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV. Hamiltonian with unbounded linear terms, J. Funct. Anal., 90 (1990), 237-283. doi: 10.1016/0022-1236(90)90084-X.

[29]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and $B$-continuous functions, J. Funct. Anal., 97 (1991), 417-465. doi: 10.1016/0022-1236(91)90010-3.

[30]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. VII. The HJB equation is not always satisfied, J. Funct. Anal., 125 (1994), 111-148. doi: 10.1006/jfan.1994.1119.

[31]

A. Fathi, "Weak KAM Theory in Lagrangian Dynamics," Cambridge Studies in Advanced Mathematics, Cambridge University Press, New York, 2012.

[32]

J. Feng, A Hamilton-Jacobi PDE in the space of measures and its associated compressible Euler equations, C. R. Math. Acad. Sci. Paris, 349 (2011), 973-976. doi: 10.1016/j.crma.2011.08.013.

[33]

J. Feng and M. Katsoulakis, A comparison principle for Hamilton-Jacobi equations related to controlled gradient flows in infinite dimensions, Arch. Ration. Mech. Anal., 192 (2009), 275-310. doi: 10.1007/s00205-008-0133-5.

[34]

J. Feng and T. Kurtz, "Large Deviations for Stochastic Processes," Mathematical Surveys and Monographs, 131, American Mathematical Society, Providence, RI, 2006.

[35]

J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, J. Math. Pures Appl. (9), 97 (2012), 318-390. doi: 10.1016/j.matpur.2011.11.004.

[36]

J. Feng and A. Świech, Optimal control for a mixed flow of Hamiltonian and gradient type in space of probability measures, With an appendix by Atanas Stefanov, Trans. Amer. Math. Soc., 365 (2013), 3987-4039. doi: 10.1090/S0002-9947-2013-05634-6.

[37]

I. Fonseca and W. Gangbo, "Degree Theory in Analysis and Its Applications," Oxford Lecture Series in Mathematics and its Applications, 2, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

[38]

W. Gangbo, H. K. Kim and T. Pacini, Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems, Memoirs of the AMS, 211 (2011). doi: 10.1090/S0065-9266-2010-00610-0.

[39]

W. Gangbo and R. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620.

[40]

W. Gangbo, T. Nguyen and A. Tudorascu, Euler-Poisson systems as action-minimizing paths in the Wasserstein space, Arch. Ration. Mech. Anal., 192 (2009), 419-452. doi: 10.1007/s00205-008-0148-y.

[41]

W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Methods and Applications of Analysis, 15 (2008), 155-183.

[42]

W. Gangbo and A. Tudorascu, Homogenization for a class of integral functionals in spaces of probability measures, Advances in Mathematics, 230 (2012), 1124-1173. doi: 10.1016/j.aim.2012.03.005.

[43]

W. Gangbo and A. Tudorascu, Weak KAM theory on the Wasserstein with multi-dimensional underlying space,, to appear in Comm. Pure Applied Math., (). 

[44]

Y. Giga, N. Hamamuki and A. Nakayasu, Eikonal equations in metric spaces,, preprint., (). 

[45]

D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308-328. doi: 10.1016/j.matpur.2009.10.010.

[46]

D. A. Gomes and L. Nurbekyan, Weak kam theory on the d-infinite dimensional torus,, preprint., (). 

[47]

D. A. Gomes and L. Nurbekyan, On the minimizers of calculus of variations problems in Hilbert spaces,, preprint., (). 

[48]

N. Gozlan, C. Roberto and P. M. Samson, Hamilton Jacobi equations on metric spaces and transport-entropy inequalities,, preprint, (). 

[49]

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

[50]

O. Guéant, J.-M. Lasry and P.-L. Lions, Application of mean field games to growth theory,, preprint., (). 

[51]

O. Guéant, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), 276-294. doi: 10.1016/j.matpur.2009.04.008.

[52]

O. Guéant, "Mean Field Games and Applications to Economics," Ph.D Thesis, Université Paris Dauphine, 2010.

[53]

, R. Hynd and H.-K. Kim,, work in progress., (). 

[54]

H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces, J. Funct. Anal., 105 (1992), 301-341. doi: 10.1016/0022-1236(92)90081-S.

[55]

B. Khesin and P. Lee, Poisson geometry and first integrals of geostrophic equations, Phys. D, 237 (2008), 2072-2077. doi: 10.1016/j.physd.2008.03.001.

[56]

J.-M. Lasry and P.-L. Lions, Large investor trading impacts on volatility, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 311-323. doi: 10.1016/j.anihpc.2005.12.006.

[57]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, (French) [Mean field games. I. The stationary case], C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.

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J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et controle optimal, (French) [Mean field games. II. Finite horizon and optimal control], C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.

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show all references

References:
[1]

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

[2]

Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109. doi: 10.1137/100790069.

[3]

G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in $ R^n $ , Math. Z., 230 (1999), 259-316. doi: 10.1007/PL00004691.

[4]

L. Ambrosio and J. Feng, On a class of first order Hamilton-Jacobi equations in metric spaces,, preprint., (). 

[5]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.

[6]

L. Ambrosio and W. Gangbo, Hamiltonian ODEs in the Wasserstein space of probability measures, Comm. Pure Appl. Math., 61 (2007), 18-53. doi: 10.1002/cpa.20188.

[7]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.

[8]

L. Ambrosio, N. Gigli and G. Savaré, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 969-996. doi: 10.4171/RMI/746.

[9]

V. Barbu and G. Da Prato, "Hamilton-Jacobi Equations in Hilbert Spaces," Research Notes in Mathematics, 86, Pitman (Advanced Publishing Program), Boston, MA, 1983.

[10]

P. Bernard and B. Buffoni, Optimal mass transportation and Mather theory, Journal of the European Mathematical Society, 9 (2007), 85-121. doi: 10.4171/JEMS/74.

[11]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Macroscopic fluctuation theory for stationary non-equilibrium states, J. Statist. Phys., 107 (2002), 635-675. doi: 10.1023/A:1014525911391.

[12]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Large deviations for the boundary driven symmetric simple exclusion process, Math. Phys. Anal. Geom., 6 (2003), 231-267. doi: 10.1023/A:1024967818899.

[13]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Minimum dissipation principle in stationary non-equilibrium states, J. Statist. Phys., 116 (2004), 831-841. doi: 10.1023/B:JOSS.0000037220.57358.94.

[14]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Stochastic interacting particle systems out of equilibrium, J. Stat. Mech. Theory Exp., 2007 (2007), P07014, 35 pp.

[15]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Action functional and quasi-potential for the Burgers equation in a bounded interval, Comm. Pure Appl. Math., 64 (2011), 649-696. doi: 10.1002/cpa.20357.

[16]

L. Bertini, D. Gabrielli and J. L. Lebowitz, Large deviations for a stochastic model of heat flow, J. Stat. Phys., 121 (2005), 843-885. doi: 10.1007/s10955-005-5527-2.

[17]

A. J. Bertozzi and A. Majda, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.

[18]

J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc., 303 (1987), 517-527. doi: 10.1090/S0002-9947-1987-0902782-7.

[19]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402.

[20]

Y. Brenier, W. Gangbo, G. Savaré and M. Westdickenberg, Sticky particle dynamics with interactions, Journal de Math. Pures et Appliquées (9), 99 (2013), 577-617. doi: 10.1016/j.matpur.2012.09.013.

[21]

P. Cardialaguet, J.-M. Lasry, P.-L. Lions and A. Porretta, Long times average of mean field games, Networks and Heterogeneous Media, 7 (2012), 279-301. doi: 10.3934/nhm.2012.7.279.

[22]

P. Cardialaguet and M. Quincampoix, Deterministic differential games under probability knowledge of initial condition, Int. Game Theory Rev., 10 (2008), 1-16. doi: 10.1142/S021919890800173X.

[23]

M. G. Crandall, H. Ishii and P.-L. Lions, User's Guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.

[24]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in Banach spaces, in "Trends in the Theory and Practice of Nonlinear Analysis" (Arlington, Tex., 1984), North-Holland Math. Stud., 110, North-Holland, Amsterdam, (1985), 115-119. doi: 10.1016/S0304-0208(08)72698-7.

[25]

M. G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. I. Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), 379-396. doi: 10.1016/0022-1236(85)90011-4.

[26]

M. G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions, J. Funct. Anal., 63 (1986), 368-405. doi: 10.1016/0022-1236(86)90026-1.

[27]

M. G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. III, J. Funct. Anal., 68 (1986), 214-247. doi: 10.1016/0022-1236(86)90005-4.

[28]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV. Hamiltonian with unbounded linear terms, J. Funct. Anal., 90 (1990), 237-283. doi: 10.1016/0022-1236(90)90084-X.

[29]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and $B$-continuous functions, J. Funct. Anal., 97 (1991), 417-465. doi: 10.1016/0022-1236(91)90010-3.

[30]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. VII. The HJB equation is not always satisfied, J. Funct. Anal., 125 (1994), 111-148. doi: 10.1006/jfan.1994.1119.

[31]

A. Fathi, "Weak KAM Theory in Lagrangian Dynamics," Cambridge Studies in Advanced Mathematics, Cambridge University Press, New York, 2012.

[32]

J. Feng, A Hamilton-Jacobi PDE in the space of measures and its associated compressible Euler equations, C. R. Math. Acad. Sci. Paris, 349 (2011), 973-976. doi: 10.1016/j.crma.2011.08.013.

[33]

J. Feng and M. Katsoulakis, A comparison principle for Hamilton-Jacobi equations related to controlled gradient flows in infinite dimensions, Arch. Ration. Mech. Anal., 192 (2009), 275-310. doi: 10.1007/s00205-008-0133-5.

[34]

J. Feng and T. Kurtz, "Large Deviations for Stochastic Processes," Mathematical Surveys and Monographs, 131, American Mathematical Society, Providence, RI, 2006.

[35]

J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, J. Math. Pures Appl. (9), 97 (2012), 318-390. doi: 10.1016/j.matpur.2011.11.004.

[36]

J. Feng and A. Świech, Optimal control for a mixed flow of Hamiltonian and gradient type in space of probability measures, With an appendix by Atanas Stefanov, Trans. Amer. Math. Soc., 365 (2013), 3987-4039. doi: 10.1090/S0002-9947-2013-05634-6.

[37]

I. Fonseca and W. Gangbo, "Degree Theory in Analysis and Its Applications," Oxford Lecture Series in Mathematics and its Applications, 2, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

[38]

W. Gangbo, H. K. Kim and T. Pacini, Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems, Memoirs of the AMS, 211 (2011). doi: 10.1090/S0065-9266-2010-00610-0.

[39]

W. Gangbo and R. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620.

[40]

W. Gangbo, T. Nguyen and A. Tudorascu, Euler-Poisson systems as action-minimizing paths in the Wasserstein space, Arch. Ration. Mech. Anal., 192 (2009), 419-452. doi: 10.1007/s00205-008-0148-y.

[41]

W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Methods and Applications of Analysis, 15 (2008), 155-183.

[42]

W. Gangbo and A. Tudorascu, Homogenization for a class of integral functionals in spaces of probability measures, Advances in Mathematics, 230 (2012), 1124-1173. doi: 10.1016/j.aim.2012.03.005.

[43]

W. Gangbo and A. Tudorascu, Weak KAM theory on the Wasserstein with multi-dimensional underlying space,, to appear in Comm. Pure Applied Math., (). 

[44]

Y. Giga, N. Hamamuki and A. Nakayasu, Eikonal equations in metric spaces,, preprint., (). 

[45]

D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308-328. doi: 10.1016/j.matpur.2009.10.010.

[46]

D. A. Gomes and L. Nurbekyan, Weak kam theory on the d-infinite dimensional torus,, preprint., (). 

[47]

D. A. Gomes and L. Nurbekyan, On the minimizers of calculus of variations problems in Hilbert spaces,, preprint., (). 

[48]

N. Gozlan, C. Roberto and P. M. Samson, Hamilton Jacobi equations on metric spaces and transport-entropy inequalities,, preprint, (). 

[49]

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

[50]

O. Guéant, J.-M. Lasry and P.-L. Lions, Application of mean field games to growth theory,, preprint., (). 

[51]

O. Guéant, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), 276-294. doi: 10.1016/j.matpur.2009.04.008.

[52]

O. Guéant, "Mean Field Games and Applications to Economics," Ph.D Thesis, Université Paris Dauphine, 2010.

[53]

, R. Hynd and H.-K. Kim,, work in progress., (). 

[54]

H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces, J. Funct. Anal., 105 (1992), 301-341. doi: 10.1016/0022-1236(92)90081-S.

[55]

B. Khesin and P. Lee, Poisson geometry and first integrals of geostrophic equations, Phys. D, 237 (2008), 2072-2077. doi: 10.1016/j.physd.2008.03.001.

[56]

J.-M. Lasry and P.-L. Lions, Large investor trading impacts on volatility, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 311-323. doi: 10.1016/j.anihpc.2005.12.006.

[57]

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