Advertisement

Entropic Ricci Curvature for Discrete Spaces

  • Jan Maas
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2184)

Abstract

We give a short overview on a recently developed notion of Ricci curvature for discrete spaces. This notion relies on geodesic convexity properties of the relative entropy along geodesics in the space of probability densities, for a metric which is similar to (but different from) the 2-Wasserstein metric. The theory can be considered as a discrete counterpart to the theory of Ricci curvature for geodesic measure spaces developed by Lott–Sturm–Villani.

References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2008)zbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163(7), 1405–1490 (2014). http://dx.doi.org/10.1215/00127094-2681605.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/1984. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985). http://dx.doi.org/10.1007/BFb0075847
  4. 4.
    Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000). http://dx.doi.org/10.1007/s002110050002 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonciocat, A.I., Sturm, K.T.: Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256(9), 2944–2966 (2009). http://dx.doi.org/10.1016/j.jfa.2009.01.029 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caputo, P., Dai Pra, P., Posta, G.: Convex entropy decay via the Bochner-Bakry-Emery approach. Ann. Inst. Henri Poincaré Probab. Stat. 45(3), 734–753 (2009). http://dx.doi.org/10.1214/08-AIHP183 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Che, R., Huang, W., Li, Y., Tetali, P.: Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities. J. Differ. Equ. 261(4), 2552–2583 (2016). http://dx.doi.org/10.1016/j.jde.2016.05.003 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chow, S.N., Huang, W., Li, Y., Zhou, H.: Fokker-Planck equations for a free energy functional or Markov process on a graph. Arch. Ration. Mech. Anal. 203(3), 969–1008 (2012). http://dx.doi.org/10.1007/s00205-011-0471-6 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146(2), 219–257 (2001). http://dx.doi.org/10.1007/s002220100160 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Erbar, M., Fathi, M.: Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature. arXiv preprint arXiv:1612.00514 (2016)Google Scholar
  11. 11.
    Erbar, M., Maas, J.: Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206(3), 997–1038 (2012). http://dx.doi.org/10.1007/s00205-012-0554-z MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Erbar, M., Maas, J.: Gradient flow structures for discrete porous medium equations. Discrete Contin. Dyn. Syst. 34(4), 1355–1374 (2014). http://dx.doi.org/10.3934/dcds.2014.34.1355 MathSciNetzbMATHGoogle Scholar
  13. 13.
    Erbar, M., Kuwada, K., Sturm, K.T.: On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Invent. Math. 201(3), 993–1071 (2015). http://dx.doi.org/10.1007/s00222-014-0563-7 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Erbar, M., Maas, J., Tetali, P.: Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models. Ann. Fac. Sci. Toulouse Math. (6) 24(4), 781–800 (2015). http://dx.doi.org/10.5802/afst.1464
  15. 15.
    Erbar, M., Henderson, C., Menz, G., Tetali, P.: Ricci curvature bounds for weakly interacting Markov chains. arXiv preprint arXiv:1602.05414 (2016)Google Scholar
  16. 16.
    Fathi, M., Maas, J.: Entropic Ricci curvature bounds for discrete interacting systems. Ann. Appl. Probab. 26(3), 1774–1806 (2016). http://dx.doi.org/10.1214/15-AAP1133 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gozlan, N., Roberto, C., Samson, P.M., Tetali, P.: Displacement convexity of entropy and related inequalities on graphs. Probab. Theory Relat. Fields 160(1–2), 47–94 (2014). http://dx.doi.org/10.1007/s00440-013-0523-y MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Léonard, C.: Lazy random walks and optimal transport on graphs. Ann. Probab. 44(3), 1864–1915 (2016). http://dx.doi.org/10.1214/15-AOP1012 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009). http://dx.doi.org/10.4007/annals.2009.169.903
  21. 21.
    Maas, J.: Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261(8), 2250–2292 (2011). http://dx.doi.org/10.1016/j.jfa.2011.06.009 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997). http://dx.doi.org/10.1006/aima.1997.1634 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mielke, A.: A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24(4), 1329–1346 (2011). http://dx.doi.org/10.1088/0951-7715/24/4/016 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mielke, A.: Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial Differ. Equ. 48(1–2), 1–31 (2013). http://dx.doi.org/10.1007/s00526-012-0538-8 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ollivier, Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256(3), 810–864 (2009). http://dx.doi.org/10.1016/j.jfa.2008.11.001 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001). http://dx.doi.org/10.1081/PDE-100002243 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000). http://dx.doi.org/10.1006/jfan.1999.3557 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Renesse, M.K.v., Sturm, K.T.: Transport inequalities, gradient estimates, entropy, and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005). http://dx.doi.org/10.1002/cpa.20060
  29. 29.
    Sturm, K.T.: On the geometry of metric measure spaces. I and II. Acta Math. 196(1), 65–177 (2006). http://dx.doi.org/10.1007/s11511-006-0003-7 MathSciNetzbMATHGoogle Scholar
  30. 30.
    Villani, C.: Optimal Transport, Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009). http://dx.doi.org/10.1007/978-3-540-71050-9

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Institute of Science and Technology Austria (IST Austria)KlosterneuburgAustria

Personalised recommendations