# 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 - A, 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. 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, (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. doi: 10.1007/s00440-008-0177-3. Google Scholar [4] P. Billingsley, Probability and Measure,, 2nd edition, (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. 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). 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). 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, (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. 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. 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. 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. doi: 10.1007/BF02179656. Google Scholar [13] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport,, in Optimal Transportation, (2014), 100. 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. 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. Google Scholar [16] N. Dirr, V. Laschos and J. Zimmer, Upscaling from particle models to entropic gradient flows,, J. Math. Phys., 53 (2012). doi: 10.1063/1.4726509. Google Scholar [17] R. Dobrushin, Vlasov equations,, Functional Analysis and Its Applications, 13 (1979), 48. 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. 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. 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, (1997). 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. 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. 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. 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. 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, (2014), 167. Google Scholar [27] N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics,, SIAM J. Math. Anal., 45 (2013), 879. 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. 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, (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. 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. 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. 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. 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. doi: 10.1007/s00526-012-0538-8. Google Scholar [35] A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems,, Springer International Publishing, 3 (2016), 187. 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. 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. 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. 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. 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, (1464), 165. doi: 10.1007/BFb0085169. Google Scholar

show all references

##### 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. 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, (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. doi: 10.1007/s00440-008-0177-3. Google Scholar [4] P. Billingsley, Probability and Measure,, 2nd edition, (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. 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). 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). 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, (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. 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. 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. 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. doi: 10.1007/BF02179656. Google Scholar [13] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport,, in Optimal Transportation, (2014), 100. 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. 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. Google Scholar [16] N. Dirr, V. Laschos and J. Zimmer, Upscaling from particle models to entropic gradient flows,, J. Math. Phys., 53 (2012). doi: 10.1063/1.4726509. Google Scholar [17] R. Dobrushin, Vlasov equations,, Functional Analysis and Its Applications, 13 (1979), 48. 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. 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. 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, (1997). 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. 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. 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. 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. 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, (2014), 167. Google Scholar [27] N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics,, SIAM J. Math. Anal., 45 (2013), 879. 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. 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, (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. 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. 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. 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. 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. doi: 10.1007/s00526-012-0538-8. Google Scholar [35] A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems,, Springer International Publishing, 3 (2016), 187. 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. 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. 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. 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. 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, (1464), 165. doi: 10.1007/BFb0085169. Google Scholar
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