American Institute of Mathematical Sciences

December  2016, 36(12): 6799-6833. doi: 10.3934/dcds.2016096

Gradient flow structure for McKean-Vlasov equations on discrete spaces

 1 University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn 2 University of California, Berkeley, Evans Hall, Berkeley, California 94720-3840, United States 3 Weierstrass Institut, Mohrenstraße 39, 10117 Berlin, Germany 4 University of Bonn, Germany, Endenicher Allee 60, 53115 Bonn, Germany

Received  January 2016 Revised  August 2016 Published  October 2016

In this work, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of $N$-particle dynamics, as $N$ goes to infinity.
Citation: Matthias Erbar, Max Fathi, Vaios Laschos, André Schlichting. Gradient flow structure for McKean-Vlasov equations on discrete spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6799-6833. doi: 10.3934/dcds.2016096
References:
 [1] S. Adams, N. Dirr, M. A. Peletier and J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage, Comm. Math. Phys., 307 (2011), 791-815. doi: 10.1007/s00220-011-1328-4.  Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. doi: 10.1007/978-3-7643-8722-8.  Google Scholar [3] L. Ambrosio, G. Savaré and L. Zambotti, Existence and stability for Fokker-Planck equations with log-concave reference measure, Probab. Theory Related Fields, 145 (2009), 517-564. doi: 10.1007/s00440-008-0177-3.  Google Scholar [4] P. Billingsley, Probability and Measure, 2nd edition, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1999.  Google Scholar [5] F. Bolley, A. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Related Fields, 137 (2007), 541-593. doi: 10.1007/s00440-006-0004-7.  Google Scholar [6] A. Budhiraja, P. Dupuis, M. Fischer and K. Ramanan, Limits of relative entropies associated with weakly interacting particle systems, Electron. J. Probab., 20 (2015), 22pp. doi: 10.1214/EJP.v20-4003.  Google Scholar [7] A. Budhiraja, P. Dupuis, M. Fischer and K. Ramanan, Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes, Electron. J. Probab., 20 (2015), 30pp. doi: 10.1214/EJP.v20-4004.  Google Scholar [8] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, vol. 207 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar [9] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376.  Google Scholar [10] J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263. doi: 10.1007/s00205-005-0386-1.  Google Scholar [11] P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case, Probab. Theory Related Fields, 140 (2008), 19-40. doi: 10.1007/s00440-007-0056-3.  Google Scholar [12] P. Dai Pra and F. den Hollander, McKean-Vlasov limit for interacting random processes in random media, J. Statist. Phys., 84 (1996), 735-772. doi: 10.1007/BF02179656.  Google Scholar [13] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, in Optimal Transportation, vol. 413 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2014, 100-144. doi: 10.1017/CBO9781107297296.007.  Google Scholar [14] D. A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308. doi: 10.1080/17442508708833446.  Google Scholar [15] E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180-187.  Google Scholar [16] N. Dirr, V. Laschos and J. Zimmer, Upscaling from particle models to entropic gradient flows, J. Math. Phys., 53 (2012), 063704, 9 pp. doi: 10.1063/1.4726509.  Google Scholar [17] R. Dobrushin, Vlasov equations, Functional Analysis and Its Applications, 13 (1979), 48-58,96. doi: 10.1007/BF01077243.  Google Scholar [18] J. Dolbeault, B. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34 (2009), 193-231. doi: 10.1007/s00526-008-0182-5.  Google Scholar [19] M. H. Duong, V. Laschos and M. Renger, Wasserstein gradient flows from large deviations of many-particle limits, ESAIM Control Optim. Calc. Var., 19 (2013), 1166-1188. doi: 10.1051/cocv/2013049.  Google Scholar [20] P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1997, A Wiley-Interscience Publication. doi: 10.1002/9781118165904.  Google Scholar [21] M. Erbar, Gradient flows of the entropy for jump processes, Ann. Inst. H. Poincaré Probab. Statist., 50 (2014), 920-945. doi: 10.1214/12-AIHP537.  Google Scholar [22] M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z.  Google Scholar [23] M. Erbar and J. Maas, Gradient flow structures for discrete porous medium equations, Discrete Contin. Dyn. Syst., 34 (2014), 1355-1374. doi: 10.3934/dcds.2014.34.1355.  Google Scholar [24] M. Erbar, J. Maas and M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Electron. Commun. Probab., 20 (2015), 1-12. doi: 10.1214/ECP.v20-4315.  Google Scholar [25] M. Fathi, A gradient flow approach to large deviations for diffusion processes, J. Math. Pures Appl., (2016). doi: 10.1016/j.matpur.2016.03.018.  Google Scholar [26] M. Fathi and M. Simon, The gradient flow approach to hydrodynamic limits for the simple exclusion process, In P. Gonçalves and A. J. Soares, From Particle Systems to Partial Differential Equations III: Particle Systems and PDEs III, Braga, Portugal, December 2014, 167-184, Springer International Publishing, Cham, 2016. Google Scholar [27] N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Anal., 45 (2013), 879-899. doi: 10.1137/120886315.  Google Scholar [28] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.  Google Scholar [29] C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, vol. 320 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-03752-2.  Google Scholar [30] D. A. Levin, M. J. Luczak and Y. Peres, Glauber dynamics for the mean-field Ising model: Cut-off, critical power law, and metastability, Probab. Theory Related Fields, 146 (2010), 223-265. doi: 10.1007/s00440-008-0189-z.  Google Scholar [31] J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009.  Google Scholar [32] F. Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes, Ann. Appl. Probab., 13 (2003), 540-560. doi: 10.1214/aoap/1050689593.  Google Scholar [33] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equations of state calculations by fast computing machines, J. Chem. Phys., 21 (1953), 1087-1091. doi: 10.1063/1.1699114.  Google Scholar [34] A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations, 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8.  Google Scholar [35] A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems, Springer International Publishing, Cham, 3 (2016), 187-249. doi: 10.1007/978-3-319-26883-5_3.  Google Scholar [36] K. Oelschläger, A martingale approach to the law of large numbers for weakly interacting stochastic processes, Ann. Probab., 12 (1984), 458-479. doi: 10.1214/aop/1176993301.  Google Scholar [37] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar [38] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.  Google Scholar [39] A. Schlichting, Macroscopic limits of the Becker-Döring equations via gradient flows,, , ().   Google Scholar [40] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427.  Google Scholar [41] A.-S. Sznitman, Topics in Propagation of Chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, vol. 1464 of Lecture Notes in Math., Springer, Berlin, 1991, 165-251. doi: 10.1007/BFb0085169.  Google Scholar

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References:
 [1] S. Adams, N. Dirr, M. A. Peletier and J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage, Comm. Math. Phys., 307 (2011), 791-815. doi: 10.1007/s00220-011-1328-4.  Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. doi: 10.1007/978-3-7643-8722-8.  Google Scholar [3] L. Ambrosio, G. Savaré and L. Zambotti, Existence and stability for Fokker-Planck equations with log-concave reference measure, Probab. Theory Related Fields, 145 (2009), 517-564. doi: 10.1007/s00440-008-0177-3.  Google Scholar [4] P. Billingsley, Probability and Measure, 2nd edition, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1999.  Google Scholar [5] F. Bolley, A. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Related Fields, 137 (2007), 541-593. doi: 10.1007/s00440-006-0004-7.  Google Scholar [6] A. Budhiraja, P. Dupuis, M. Fischer and K. Ramanan, Limits of relative entropies associated with weakly interacting particle systems, Electron. J. Probab., 20 (2015), 22pp. doi: 10.1214/EJP.v20-4003.  Google Scholar [7] A. Budhiraja, P. Dupuis, M. Fischer and K. Ramanan, Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes, Electron. J. Probab., 20 (2015), 30pp. doi: 10.1214/EJP.v20-4004.  Google Scholar [8] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, vol. 207 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar [9] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376.  Google Scholar [10] J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263. doi: 10.1007/s00205-005-0386-1.  Google Scholar [11] P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case, Probab. Theory Related Fields, 140 (2008), 19-40. doi: 10.1007/s00440-007-0056-3.  Google Scholar [12] P. Dai Pra and F. den Hollander, McKean-Vlasov limit for interacting random processes in random media, J. Statist. Phys., 84 (1996), 735-772. doi: 10.1007/BF02179656.  Google Scholar [13] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, in Optimal Transportation, vol. 413 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2014, 100-144. doi: 10.1017/CBO9781107297296.007.  Google Scholar [14] D. A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308. doi: 10.1080/17442508708833446.  Google Scholar [15] E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180-187.  Google Scholar [16] N. Dirr, V. Laschos and J. Zimmer, Upscaling from particle models to entropic gradient flows, J. Math. Phys., 53 (2012), 063704, 9 pp. doi: 10.1063/1.4726509.  Google Scholar [17] R. Dobrushin, Vlasov equations, Functional Analysis and Its Applications, 13 (1979), 48-58,96. doi: 10.1007/BF01077243.  Google Scholar [18] J. Dolbeault, B. Nazaret and G. Savaré, A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34 (2009), 193-231. doi: 10.1007/s00526-008-0182-5.  Google Scholar [19] M. H. Duong, V. Laschos and M. Renger, Wasserstein gradient flows from large deviations of many-particle limits, ESAIM Control Optim. Calc. Var., 19 (2013), 1166-1188. doi: 10.1051/cocv/2013049.  Google Scholar [20] P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1997, A Wiley-Interscience Publication. doi: 10.1002/9781118165904.  Google Scholar [21] M. Erbar, Gradient flows of the entropy for jump processes, Ann. Inst. H. Poincaré Probab. Statist., 50 (2014), 920-945. doi: 10.1214/12-AIHP537.  Google Scholar [22] M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z.  Google Scholar [23] M. Erbar and J. Maas, Gradient flow structures for discrete porous medium equations, Discrete Contin. Dyn. Syst., 34 (2014), 1355-1374. doi: 10.3934/dcds.2014.34.1355.  Google Scholar [24] M. Erbar, J. Maas and M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Electron. Commun. Probab., 20 (2015), 1-12. doi: 10.1214/ECP.v20-4315.  Google Scholar [25] M. Fathi, A gradient flow approach to large deviations for diffusion processes, J. Math. Pures Appl., (2016). doi: 10.1016/j.matpur.2016.03.018.  Google Scholar [26] M. Fathi and M. Simon, The gradient flow approach to hydrodynamic limits for the simple exclusion process, In P. Gonçalves and A. J. Soares, From Particle Systems to Partial Differential Equations III: Particle Systems and PDEs III, Braga, Portugal, December 2014, 167-184, Springer International Publishing, Cham, 2016. Google Scholar [27] N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Anal., 45 (2013), 879-899. doi: 10.1137/120886315.  Google Scholar [28] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.  Google Scholar [29] C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, vol. 320 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-03752-2.  Google Scholar [30] D. A. Levin, M. J. Luczak and Y. Peres, Glauber dynamics for the mean-field Ising model: Cut-off, critical power law, and metastability, Probab. Theory Related Fields, 146 (2010), 223-265. doi: 10.1007/s00440-008-0189-z.  Google Scholar [31] J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009.  Google Scholar [32] F. Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes, Ann. Appl. Probab., 13 (2003), 540-560. doi: 10.1214/aoap/1050689593.  Google Scholar [33] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equations of state calculations by fast computing machines, J. Chem. Phys., 21 (1953), 1087-1091. doi: 10.1063/1.1699114.  Google Scholar [34] A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations, 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8.  Google Scholar [35] A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems, Springer International Publishing, Cham, 3 (2016), 187-249. doi: 10.1007/978-3-319-26883-5_3.  Google Scholar [36] K. Oelschläger, A martingale approach to the law of large numbers for weakly interacting stochastic processes, Ann. Probab., 12 (1984), 458-479. doi: 10.1214/aop/1176993301.  Google Scholar [37] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar [38] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.  Google Scholar [39] A. Schlichting, Macroscopic limits of the Becker-Döring equations via gradient flows,, , ().   Google Scholar [40] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427.  Google Scholar [41] A.-S. Sznitman, Topics in Propagation of Chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, vol. 1464 of Lecture Notes in Math., Springer, Berlin, 1991, 165-251. doi: 10.1007/BFb0085169.  Google Scholar
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