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Archive for Rational Mechanics and Analysis

, Volume 206, Issue 3, pp 997–1038 | Cite as

Ricci Curvature of Finite Markov Chains via Convexity of the Entropy

  • Matthias Erbar
  • Jan Maas
Article

Abstract

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry–Émery and Otto–Villani. Further, we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.

Keywords

Entropy Markov Chain Ricci Curvature Discrete Analogue Simple Random Walk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel, 2008Google Scholar
  2. 2.
    Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Preprint at arXiv:1106.2090, 2011Google Scholar
  3. 3.
    Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Preprint at arXiv:1109.0222, 2011Google Scholar
  4. 4.
    Ané C., Ledoux M.: On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Relat. Fields 116(4), 573–602 (2000)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bakry, D., Émery, M.: Diffusions hypercontractives. Séminaire de probabilités, XIX, 1983/84. Lecture Notes in Math., vol. 1123. Springer, Berlin, 177–206, 1985Google Scholar
  6. 6.
    Bauer, F., Jost, J., Liu, S.: Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator. Preprint at arXiv:1105.3803, 2011Google Scholar
  7. 7.
    Benamou J.-D., Brenier Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bobkov S.G., Götze F.: Discrete isoperimetric and Poincaré-type inequalities. Probab. Theory Relat. Fields 114(2), 245–277 (1999)zbMATHCrossRefGoogle Scholar
  9. 9.
    Bobkov S.G., Houdré C., Tetali P.: The subgaussian constant and concentration inequalities. Isr. J. Math. 156, 255–283 (2006)zbMATHCrossRefGoogle Scholar
  10. 10.
    Bobkov S.G., Ledoux M.: On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156(2), 347–365 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bobkov S.G., Tetali P.: Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19(2), 289–336 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bonciocat A.-I., Sturm K.-Th.: Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256(9), 2944–2966 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Buttazzo, G.: Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Research Notes in Mathematics Series. Longman Scientific and Technical, Harlow, 1989Google Scholar
  14. 14.
    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)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Chow S.-N., Huang W., Li Y., Zhou H.: Fokker-Planck equations for a free energy functional or Markov process on a graph. Arch. Rational Mech. Anal. 203, 969–1008 (2012)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Daneri S., Savaré G.: Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Anal. 40(3), 1104–1122 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Dolbeault J., Nazaret B., Savaré G.: A new class of transport distances between measures. Calc. Var. Partial Differ. Equ. 34(2), 193–231 (2009)zbMATHCrossRefGoogle Scholar
  18. 18.
    Erbar M.: The heat equation on manifolds as a gradient flow in the Wasserstein space. Ann. Inst. Henri Poincaré Probab. Stat. 46(1), 1–23 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Fang S., Shao J., Sturm K.-Th.: Wasserstein space over the Wiener space. Probab. Theory Relat. Fields 146(3–4), 535–565 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gigli, N., Kuwada, K.: Ohta. Heat flow on Alexandrov spaces. Preprint at arXiv:1008.1319, 2010Google Scholar
  21. 21.
    Gozlan N., Léonard C.: Transport inequalities. A survey. Markov Process. Relat. Fields 16(4), 635–736 (2010)zbMATHGoogle Scholar
  22. 22.
    Hua, B., Jost, J., Liu, S.: Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature. Preprint at arXiv:1107.2826, 2011Google Scholar
  23. 23.
    Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Jost, J., Liu, S.: Ollivier’s Ricci curvature, local clustering and curvature dimension inequalities on graphs. Preprint at arXiv:1103.4037, 2011Google Scholar
  25. 25.
    Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence, 2001Google Scholar
  26. 26.
    Lin Y., Yau S.-T.: Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett. 17(2), 343–356 (2010)MathSciNetADSzbMATHGoogle Scholar
  27. 27.
    Lott J., Villani C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Maas J.: Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261(8), 2250–2292 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    McCann R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Mielke, A.: Geodesic convexity of the relative entropy in reversible Markov chains. Preprint, 2011Google Scholar
  31. 31.
    Mielke A.: A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24(4), 1329–1346 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. 32.
    Ohta S.-I., Sturm K.-Th.: Heat flow on Finsler manifolds. Commun. Pure Appl. Math. 62(10), 1386–1433 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Ollivier Y.: Ricci curvature of metric spaces. C. R. Math. Acad. Sci. Paris 345(11), 643–646 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Ollivier Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256(3), 810–864 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Ollivier, Y., Villani, C.: A curved Brunn-Minkowski inequality on the discrete hypercube. Preprint at arXiv:1011.4779, 2010Google Scholar
  36. 36.
    Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Otto F., Westdickenberg M.: Eulerian calculus for the contraction in the Wasserstein distance. SIAM J. Math. Anal. 37(4), 1227–1255 (2005)MathSciNetCrossRefGoogle Scholar
  39. 39.
    von Renesse M.-K., Sturm K.-Th.: Transport inequalities, gradient estimates, entropy, and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005)zbMATHCrossRefGoogle Scholar
  40. 40.
    Sammer M., Tetali P.: Concentration on the discrete torus using transportation. Combin. Probab. Comput. 18(5), 835–860 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Sturm K.-Th.: On the geometry of metric measure spaces. I and II. Acta Math. 196(1), 65–177 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, 2003Google Scholar
  43. 43.
    Villani C.: Optimal transport, Old and new. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin, 2009Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute for Applied MathematicsUniversity of BonnBonnGermany

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