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 & Continuous Dynamical Systems - A, 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. doi: 10.1137/090758477. Google Scholar

[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. doi: 10.1137/100790069. Google Scholar

[3]

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

[4]

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

[5]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000). Google Scholar

[6]

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

[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, (2005). Google Scholar

[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. doi: 10.4171/RMI/746. Google Scholar

[9]

V. Barbu and G. Da Prato, "Hamilton-Jacobi Equations in Hilbert Spaces,", Research Notes in Mathematics, 86 (1983). Google Scholar

[10]

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

[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. doi: 10.1023/A:1014525911391. Google Scholar

[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. doi: 10.1023/A:1024967818899. Google Scholar

[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. doi: 10.1023/B:JOSS.0000037220.57358.94. Google Scholar

[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). Google Scholar

[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. doi: 10.1002/cpa.20357. Google Scholar

[16]

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

[17]

A. J. Bertozzi and A. Majda, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, 27 (2002). Google Scholar

[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. doi: 10.1090/S0002-9947-1987-0902782-7. Google Scholar

[19]

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

[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. doi: 10.1016/j.matpur.2012.09.013. Google Scholar

[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. doi: 10.3934/nhm.2012.7.279. Google Scholar

[22]

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

[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. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar

[24]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in Banach spaces,, in, 110 (1985), 115. doi: 10.1016/S0304-0208(08)72698-7. Google Scholar

[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. doi: 10.1016/0022-1236(85)90011-4. Google Scholar

[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. doi: 10.1016/0022-1236(86)90026-1. Google Scholar

[27]

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

[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. doi: 10.1016/0022-1236(90)90084-X. Google Scholar

[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. doi: 10.1016/0022-1236(91)90010-3. Google Scholar

[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. doi: 10.1006/jfan.1994.1119. Google Scholar

[31]

A. Fathi, "Weak KAM Theory in Lagrangian Dynamics,", Cambridge Studies in Advanced Mathematics, (2012). Google Scholar

[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. doi: 10.1016/j.crma.2011.08.013. Google Scholar

[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. doi: 10.1007/s00205-008-0133-5. Google Scholar

[34]

J. Feng and T. Kurtz, "Large Deviations for Stochastic Processes,", Mathematical Surveys and Monographs, 131 (2006). Google Scholar

[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. doi: 10.1016/j.matpur.2011.11.004. Google Scholar

[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, 365 (2013), 3987. doi: 10.1090/S0002-9947-2013-05634-6. Google Scholar

[37]

I. Fonseca and W. Gangbo, "Degree Theory in Analysis and Its Applications,", Oxford Lecture Series in Mathematics and its Applications, 2 (1995). Google Scholar

[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. Google Scholar

[39]

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

[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. doi: 10.1007/s00205-008-0148-y. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[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. doi: 10.1016/j.matpur.2009.10.010. Google Scholar

[46]

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

[47]

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

[48]

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

[49]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in, (2003), 205. doi: 10.1007/978-3-642-14660-2_3. Google Scholar

[50]

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

[51]

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

[52]

O. Guéant, "Mean Field Games and Applications to Economics,", Ph.D Thesis, (2010). Google Scholar

[53]

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

[54]

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

[55]

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

[56]

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

[57]

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

[58]

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], 343 (2006), 679. doi: 10.1016/j.crma.2006.09.018. Google Scholar

[59]

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

[60]

P.-L. Lions, Cours au Collège de France., Available from: , (). Google Scholar

[61]

G. Loeper, "Applications de l'Équation de Monge-Ampère à la Modélisation des Fluides et des Plasmas,", Thesis dissertation, (). Google Scholar

[62]

G. Loeper, A fully nonlinear version of the incompressible Euler equations: The semi-geostrophic system,, SIAM Journal on Mathematical Analysis, 38 (2006), 795. doi: 10.1137/050629070. Google Scholar

[63]

J. Lott, Some geometric calculations on Wasserstein space,, Comm. Math. Phys., 277 (2008), 423. doi: 10.1007/s00220-007-0367-3. Google Scholar

[64]

J. Lott and C. Villani, Hamilton-Jacobi semigroup on length spaces and applications,, J. Math. Pures Appl. (9), 88 (2007), 219. doi: 10.1016/j.matpur.2007.06.003. Google Scholar

[65]

D. Tataru, Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms,, J. Math. Pures Appl., 163 (1992), 345. doi: 10.1016/0022-247X(92)90256-D. Google Scholar

[66]

D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: A simplified approach,, J. Differential Equations, 111 (1994), 123. doi: 10.1006/jdeq.1994.1078. Google Scholar

[67]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003). doi: 10.1007/b12016. Google Scholar

[68]

V. Judovič, Non-stationary flows of an ideal incompressible fluid,, Žh. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032. Google Scholar

show all references

References:
[1]

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

[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. doi: 10.1137/100790069. Google Scholar

[3]

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

[4]

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

[5]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000). Google Scholar

[6]

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

[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, (2005). Google Scholar

[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. doi: 10.4171/RMI/746. Google Scholar

[9]

V. Barbu and G. Da Prato, "Hamilton-Jacobi Equations in Hilbert Spaces,", Research Notes in Mathematics, 86 (1983). Google Scholar

[10]

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

[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. doi: 10.1023/A:1014525911391. Google Scholar

[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. doi: 10.1023/A:1024967818899. Google Scholar

[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. doi: 10.1023/B:JOSS.0000037220.57358.94. Google Scholar

[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). Google Scholar

[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. doi: 10.1002/cpa.20357. Google Scholar

[16]

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

[17]

A. J. Bertozzi and A. Majda, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, 27 (2002). Google Scholar

[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. doi: 10.1090/S0002-9947-1987-0902782-7. Google Scholar

[19]

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

[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. doi: 10.1016/j.matpur.2012.09.013. Google Scholar

[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. doi: 10.3934/nhm.2012.7.279. Google Scholar

[22]

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

[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. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar

[24]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in Banach spaces,, in, 110 (1985), 115. doi: 10.1016/S0304-0208(08)72698-7. Google Scholar

[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. doi: 10.1016/0022-1236(85)90011-4. Google Scholar

[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. doi: 10.1016/0022-1236(86)90026-1. Google Scholar

[27]

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

[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. doi: 10.1016/0022-1236(90)90084-X. Google Scholar

[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. doi: 10.1016/0022-1236(91)90010-3. Google Scholar

[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. doi: 10.1006/jfan.1994.1119. Google Scholar

[31]

A. Fathi, "Weak KAM Theory in Lagrangian Dynamics,", Cambridge Studies in Advanced Mathematics, (2012). Google Scholar

[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. doi: 10.1016/j.crma.2011.08.013. Google Scholar

[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. doi: 10.1007/s00205-008-0133-5. Google Scholar

[34]

J. Feng and T. Kurtz, "Large Deviations for Stochastic Processes,", Mathematical Surveys and Monographs, 131 (2006). Google Scholar

[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. doi: 10.1016/j.matpur.2011.11.004. Google Scholar

[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, 365 (2013), 3987. doi: 10.1090/S0002-9947-2013-05634-6. Google Scholar

[37]

I. Fonseca and W. Gangbo, "Degree Theory in Analysis and Its Applications,", Oxford Lecture Series in Mathematics and its Applications, 2 (1995). Google Scholar

[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. Google Scholar

[39]

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

[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. doi: 10.1007/s00205-008-0148-y. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[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. doi: 10.1016/j.matpur.2009.10.010. Google Scholar

[46]

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

[47]

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

[48]

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

[49]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in, (2003), 205. doi: 10.1007/978-3-642-14660-2_3. Google Scholar

[50]

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

[51]

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

[52]

O. Guéant, "Mean Field Games and Applications to Economics,", Ph.D Thesis, (2010). Google Scholar

[53]

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

[54]

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

[55]

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

[56]

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

[57]

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

[58]

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], 343 (2006), 679. doi: 10.1016/j.crma.2006.09.018. Google Scholar

[59]

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

[60]

P.-L. Lions, Cours au Collège de France., Available from: , (). Google Scholar

[61]

G. Loeper, "Applications de l'Équation de Monge-Ampère à la Modélisation des Fluides et des Plasmas,", Thesis dissertation, (). Google Scholar

[62]

G. Loeper, A fully nonlinear version of the incompressible Euler equations: The semi-geostrophic system,, SIAM Journal on Mathematical Analysis, 38 (2006), 795. doi: 10.1137/050629070. Google Scholar

[63]

J. Lott, Some geometric calculations on Wasserstein space,, Comm. Math. Phys., 277 (2008), 423. doi: 10.1007/s00220-007-0367-3. Google Scholar

[64]

J. Lott and C. Villani, Hamilton-Jacobi semigroup on length spaces and applications,, J. Math. Pures Appl. (9), 88 (2007), 219. doi: 10.1016/j.matpur.2007.06.003. Google Scholar

[65]

D. Tataru, Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms,, J. Math. Pures Appl., 163 (1992), 345. doi: 10.1016/0022-247X(92)90256-D. Google Scholar

[66]

D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: A simplified approach,, J. Differential Equations, 111 (1994), 123. doi: 10.1006/jdeq.1994.1078. Google Scholar

[67]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003). doi: 10.1007/b12016. Google Scholar

[68]

V. Judovič, Non-stationary flows of an ideal incompressible fluid,, Žh. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032. Google Scholar

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