Entropy dissipation of Fokker-Planck equations on graphs

Shui-Nee Chow; Wuchen Li; Haomin Zhou

doi:10.3934/dcds.2018215

Article Contents

2018, Volume 38, Issue 10: 4929-4950. Doi: 10.3934/dcds.2018215

This issue Previous Article New characterizations of Ricci curvature on RCD metric measure spaces Next Article A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations

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
* Corresponding author: Wuchen Li

Received: September 2017

Revised: January 2018

Published: October 2018

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

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Optimal transport,
entropy dissipation.

Mathematics Subject Classification: Primary: 37C10; Secondary: 39B72.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[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.
[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.
[4]	J. A. Carrillo, R. 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.
[5]	R. Che, W. Huang, Y. 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.
[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.
[7]	S.-N. Chow, W. Huang, Y. 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.
[8]	S.-N. Chow, W. Li, J. Lu and H. Zhou, Population games and discrete optimal transport, arXiv: 1704.00855, 2017.
[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.
[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.
[11]	M. Erbar, M. Fathi, V. 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.
[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.
[13]	M. Erbar and J. Maas, Gradient flow structures for discrete porous medium equations, arXiv: 1212.1129, 2012.
[14]	M. Erbar, K. 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.
[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.
[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.
[17]	R. Jordan, D. 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.
[18]	W. Li, A study of stochastic differential equations and Fokker-Planck equations with applications, Phd thesis, 2016.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	J. Solomon, R. Rustamov, L. Guibas and A. Butscher, Continuous-flow graph transportation distances, arXiv: 1603.06927, 2016.
[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.
[28]	N. G. Trillos and D. Slepcev, A variational approach to the consistency of spectral clustering, Applied and Computational Harmonic Analysis, 2016.
[29]	N. G. Trillos, Gromov-Hausdorff limit of Wasserstein spaces on point clouds, arXiv: 1702.03464, 2017.
[30]	C. Villani, Topics in Optimal Transportation, Number 58. American Mathematical Soc., 2003. doi: 10.1007/b12016.
[31]	C. Villani, Optimal Transport: Old and New, Volume 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
[32]	K. Yano, Some remarks on tensor fields and curvature, Annals of Mathematics, 55 (1952), 328-347. doi: 10.2307/1969782.
[33]	K. Yano, Some integral formulas and their applications, The Michigan Mathematical Journal, 5 (1958), 63-73. doi: 10.1307/mmj/1028998011.