October  2018, 38(10): 4929-4950. doi: 10.3934/dcds.2018215

Entropy dissipation of Fokker-Planck equations on graphs

1. 

School of Mathematics, Georgia institute of technology, Atlanta, USA

2. 

Department of Mathematics, University of California, Los Angeles, USA

* Corresponding author: Wuchen Li

Received  September 2017 Revised  January 2018 Published  July 2018

Fund Project: This work is partially supported by NSF Awards DMS-1419027, DMS-1620345, and ONR Award N000141310408

We study the nonlinear Fokker-Planck equation on graphs, which is the gradient flow in the space of probability measures supported on the nodes with respect to the discrete Wasserstein metric. The energy functional driving the gradient flow consists of a Boltzmann entropy, a linear potential and a quadratic interaction energy. We show that the solution converges to the Gibbs measures exponentially fast. The continuous analog of this asymptotic rate is related to the Yano's formula.

Citation: Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH ${{\text{Z}}^{\overset{´}{\mathop{, }}}}$urich. Birk${{\text{h}}^{\overset{´}{\mathop{, }}}} $auser Verlag, Basel, 2005. Google Scholar

[2]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the MongeKantorovich mass transfer problem, Numerische Mathematik, 84 (2000), 375-393. doi: 10.1007/s002110050002. Google Scholar

[3]

E. Carlen and W. Gangbo, Constrained steepest descent in the 2-Wasserstein metric, Annals of Mathematics, 157 (2003), 807-846. doi: 10.4007/annals.2003.157.807. Google Scholar

[4]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Revista Matematica Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376. Google Scholar

[5]

R. CheW. HuangY. Li and P. Tetali, Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities, Journal of Differential Equations, 261 (2016), 2552-2583. doi: 10.1016/j.jde.2016.05.003. Google Scholar

[6]

S.-N. Chow, L. Dieci, W. Li and H. Zhou, Entropy dissipation semi-discretization schemes for Fokker-Planck equations, Journal of Dynamics and Differential Equations, (2018), 1-28, arXiv: 1608.02628, 2016. doi: 10.1007/s10884-018-9659-x. Google Scholar

[7]

S.-N. ChowW. HuangY. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Archive for Rational Mechanics and Analysis, 203 (2012), 969-1008. doi: 10.1007/s00205-011-0471-6. Google Scholar

[8]

S.-N. Chow, W. Li, J. Lu and H. Zhou, Population games and discrete optimal transport, arXiv: 1704.00855, 2017.Google Scholar

[9]

L. Desvillettes and C. Villani, Entropic methods for the study of the long time behavior of kinetic equations, Transport Theory and Statistical Physics, 30 (2001), 155-168. doi: 10.1081/TT-100105366. Google Scholar

[10]

M. Erbar, Gradient flow of the entropy for jump processes, Ann. Inst. H. Poincare Probab. Statist, 50 (2014), 920-945. doi: 10.1214/12-AIHP537. Google Scholar

[11]

M. ErbarM. FathiV. Laschos and A. Schlichting, Gradient flow structure for McKeanVlasov equations on discrete spaces, Discrete and Continuous Dynamical System, 36 (2016), 6799-6833. doi: 10.3934/dcds.2016096. Google Scholar

[12]

M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Archive for Rational Mechanics and Analysis, 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z. Google Scholar

[13]

M. Erbar and J. Maas, Gradient flow structures for discrete porous medium equations, arXiv: 1212.1129, 2012.Google Scholar

[14]

M. ErbarK. Kuwada and K.-T. Sturm, On the equivalence of the entropic curvaturedimension condition and Bochner's inequality on metric measure spaces, Inventiones Mathematicae, 201 (2015), 993-1071. doi: 10.1007/s00222-014-0563-7. Google Scholar

[15]

M. Fathi and J. Maas, Entropic Ricci curvature bounds for discrete interacting systems, Ann. Appl. Probab., 26 (2016), 1774-1806, arXiv: 1501.00562. doi: 10.1214/15-AAP1133. Google Scholar

[16]

N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM Journal on Mathematical Analysis, 45 (2013), 879-899. doi: 10.1137/120886315. Google Scholar

[17]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359. Google Scholar

[18]

W. Li, A study of stochastic differential equations and Fokker-Planck equations with applications, Phd thesis, 2016.Google Scholar

[19]

J. Maas, Gradient flows of the entropy for finite Markov chains, Journal of Functional Analysis, 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009. Google Scholar

[20]

J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, Nonlinearity, 29 (2016), 1992-2023, arXiv: 1505.03178. doi: 10.1088/0951-7715/29/7/1992. Google Scholar

[21]

P. A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis, Mat. Contemp, 19 (2000), 1-29. Google Scholar

[22]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Dedicated to Herbert Gajewski on the Occasion of his 70th Birthday. Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016. Google Scholar

[23]

A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calculus of Variations and Partial Differential Equations, 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8. Google Scholar

[24]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Communications in Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. Google Scholar

[25]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, Journal of Functional Analysis, 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557. Google Scholar

[26]

J. Solomon, R. Rustamov, L. Guibas and A. Butscher, Continuous-flow graph transportation distances, arXiv: 1603.06927, 2016.Google Scholar

[27]

N. G. Trillos and D. Slepcev, Continuum limit of total variation on point clouds, Archive for Rational Mechanics and Analysis, 220 (2016), 193-241. doi: 10.1007/s00205-015-0929-z. Google Scholar

[28]

N. G. Trillos and D. Slepcev, A variational approach to the consistency of spectral clustering, Applied and Computational Harmonic Analysis, 2016.Google Scholar

[29]

N. G. Trillos, Gromov-Hausdorff limit of Wasserstein spaces on point clouds, arXiv: 1702.03464, 2017.Google Scholar

[30]

C. Villani, Topics in Optimal Transportation, Number 58. American Mathematical Soc., 2003. doi: 10.1007/b12016. Google Scholar

[31]

C. Villani, Optimal Transport: Old and New, Volume 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. Google Scholar

[32]

K. Yano, Some remarks on tensor fields and curvature, Annals of Mathematics, 55 (1952), 328-347. doi: 10.2307/1969782. Google Scholar

[33]

K. Yano, Some integral formulas and their applications, The Michigan Mathematical Journal, 5 (1958), 63-73. doi: 10.1307/mmj/1028998011. Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH ${{\text{Z}}^{\overset{´}{\mathop{, }}}}$urich. Birk${{\text{h}}^{\overset{´}{\mathop{, }}}} $auser Verlag, Basel, 2005. Google Scholar

[2]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the MongeKantorovich mass transfer problem, Numerische Mathematik, 84 (2000), 375-393. doi: 10.1007/s002110050002. Google Scholar

[3]

E. Carlen and W. Gangbo, Constrained steepest descent in the 2-Wasserstein metric, Annals of Mathematics, 157 (2003), 807-846. doi: 10.4007/annals.2003.157.807. Google Scholar

[4]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Revista Matematica Iberoamericana, 19 (2003), 971-1018. doi: 10.4171/RMI/376. Google Scholar

[5]

R. CheW. HuangY. Li and P. Tetali, Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities, Journal of Differential Equations, 261 (2016), 2552-2583. doi: 10.1016/j.jde.2016.05.003. Google Scholar

[6]

S.-N. Chow, L. Dieci, W. Li and H. Zhou, Entropy dissipation semi-discretization schemes for Fokker-Planck equations, Journal of Dynamics and Differential Equations, (2018), 1-28, arXiv: 1608.02628, 2016. doi: 10.1007/s10884-018-9659-x. Google Scholar

[7]

S.-N. ChowW. HuangY. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Archive for Rational Mechanics and Analysis, 203 (2012), 969-1008. doi: 10.1007/s00205-011-0471-6. Google Scholar

[8]

S.-N. Chow, W. Li, J. Lu and H. Zhou, Population games and discrete optimal transport, arXiv: 1704.00855, 2017.Google Scholar

[9]

L. Desvillettes and C. Villani, Entropic methods for the study of the long time behavior of kinetic equations, Transport Theory and Statistical Physics, 30 (2001), 155-168. doi: 10.1081/TT-100105366. Google Scholar

[10]

M. Erbar, Gradient flow of the entropy for jump processes, Ann. Inst. H. Poincare Probab. Statist, 50 (2014), 920-945. doi: 10.1214/12-AIHP537. Google Scholar

[11]

M. ErbarM. FathiV. Laschos and A. Schlichting, Gradient flow structure for McKeanVlasov equations on discrete spaces, Discrete and Continuous Dynamical System, 36 (2016), 6799-6833. doi: 10.3934/dcds.2016096. Google Scholar

[12]

M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Archive for Rational Mechanics and Analysis, 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z. Google Scholar

[13]

M. Erbar and J. Maas, Gradient flow structures for discrete porous medium equations, arXiv: 1212.1129, 2012.Google Scholar

[14]

M. ErbarK. Kuwada and K.-T. Sturm, On the equivalence of the entropic curvaturedimension condition and Bochner's inequality on metric measure spaces, Inventiones Mathematicae, 201 (2015), 993-1071. doi: 10.1007/s00222-014-0563-7. Google Scholar

[15]

M. Fathi and J. Maas, Entropic Ricci curvature bounds for discrete interacting systems, Ann. Appl. Probab., 26 (2016), 1774-1806, arXiv: 1501.00562. doi: 10.1214/15-AAP1133. Google Scholar

[16]

N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM Journal on Mathematical Analysis, 45 (2013), 879-899. doi: 10.1137/120886315. Google Scholar

[17]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359. Google Scholar

[18]

W. Li, A study of stochastic differential equations and Fokker-Planck equations with applications, Phd thesis, 2016.Google Scholar

[19]

J. Maas, Gradient flows of the entropy for finite Markov chains, Journal of Functional Analysis, 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009. Google Scholar

[20]

J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, Nonlinearity, 29 (2016), 1992-2023, arXiv: 1505.03178. doi: 10.1088/0951-7715/29/7/1992. Google Scholar

[21]

P. A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis, Mat. Contemp, 19 (2000), 1-29. Google Scholar

[22]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Dedicated to Herbert Gajewski on the Occasion of his 70th Birthday. Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016. Google Scholar

[23]

A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calculus of Variations and Partial Differential Equations, 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8. Google Scholar

[24]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Communications in Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. Google Scholar

[25]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, Journal of Functional Analysis, 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557. Google Scholar

[26]

J. Solomon, R. Rustamov, L. Guibas and A. Butscher, Continuous-flow graph transportation distances, arXiv: 1603.06927, 2016.Google Scholar

[27]

N. G. Trillos and D. Slepcev, Continuum limit of total variation on point clouds, Archive for Rational Mechanics and Analysis, 220 (2016), 193-241. doi: 10.1007/s00205-015-0929-z. Google Scholar

[28]

N. G. Trillos and D. Slepcev, A variational approach to the consistency of spectral clustering, Applied and Computational Harmonic Analysis, 2016.Google Scholar

[29]

N. G. Trillos, Gromov-Hausdorff limit of Wasserstein spaces on point clouds, arXiv: 1702.03464, 2017.Google Scholar

[30]

C. Villani, Topics in Optimal Transportation, Number 58. American Mathematical Soc., 2003. doi: 10.1007/b12016. Google Scholar

[31]

C. Villani, Optimal Transport: Old and New, Volume 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. Google Scholar

[32]

K. Yano, Some remarks on tensor fields and curvature, Annals of Mathematics, 55 (1952), 328-347. doi: 10.2307/1969782. Google Scholar

[33]

K. Yano, Some integral formulas and their applications, The Michigan Mathematical Journal, 5 (1958), 63-73. doi: 10.1307/mmj/1028998011. Google Scholar

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