Probability Theory and Related Fields

, Volume 175, Issue 3–4, pp 897–936 | Cite as

Super-Ricci flows and improved gradient and transport estimates

  • Eva KopferEmail author


We introduce Brownian motions on time-dependent metric measure spaces, proving their existence and uniqueness. We prove contraction estimates for their trajectories assuming that the time-dependent heat flow satisfies transport estimates with respect to every \(L^p\)-Kantorovich distance, \(p\in [1,\infty ]\). These transport estimates turn out to characterize super-Ricci flows, introduced by Sturm (J Funct Anal 275(12):3504–3569, 2015.)

Mathematics Subject Classification

30L99 53C44 60J25 



  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195(2), 289–391 (2013)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Gigli, N., Savaré, G.: Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43(1), 339–404 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arnaudon, M., Coulibaly, K.A., Thalmaier, A.: Horizontal diffusion in \(C^1\) path space. In: Séminaire de Probabilités XLIII, volume 2006 of Lecture Notes in Mathematics, pp. 73–94. Springer, Berlin (2011)Google Scholar
  5. 5.
    Bacher, K., Sturm, K.-T.: Ricci Bounds for Euclidean and Spherical Cones. Singular Phenomena and Scaling in Mathematical Models, pp. 3–23. Springer, Cham (2014)CrossRefGoogle Scholar
  6. 6.
    Bakry, D.: Transformations de Riesz pour les semi-groupes symétriques. II. étude sous la condition \(\Gamma _2\ge 0\). In: Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Mathematics, pp. 145–174. Springer, Berlin (1985)Google Scholar
  7. 7.
    Bauer, H.: Probability theory and elements of measure theory. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1981. Second edition of the translation by R. B. Burckel from the third German edition, Probability and Mathematical StatisticsGoogle Scholar
  8. 8.
    Bauer, H.: Wahrscheinlichkeitstheorie. de Gruyter Lehrbuch. [de Gruyter Textbook]. Walter de Gruyter & Co., Berlin, 5th edn (2002)Google Scholar
  9. 9.
    Bogachev, V.: Measure Theory, vol. 1. Springer, Berlin (2007)CrossRefGoogle Scholar
  10. 10.
    Bolley, F., Gentil, I., Guillin, A., Kuwada, K.: Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition. Ann. Sc. Norm. Super. Pisa. Cl. Sci. (5) 18(3), 845–880 (2018)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. London Mathematical Society Monographs Series, vol. 35. Princeton University Press, Princeton (2012)Google 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, 993–1071 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nicola Gigli. Nonsmooth differential geometry-An approach tailored for spaces with Ricci curvature bounded from below. arXiv:1407.0809 (2014)
  15. 15.
    Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Am. Math. Soc. 236(1113), vi–91 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Haslhofer, R., Naber, A.: Weak solutions for the Ricci flow I. arXiv:1504.00911 (2015)
  18. 18.
    Kopfer, E., Sturm, K.-T.: Heat flows on time-dependent metric measure spaces and super-Ricci Flows. Commun. Pure Appl. Math. arXiv:1611.02570 (2017)
  19. 19.
    Kuwada, K.: Duality on gradient estimates and Wasserstein controls. J. Funct. Anal. 258(11), 3758–3774 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kuwada, K., Philipowski, R.: Coupling of Brownian motions and Perelman’s L-functional. J. Funct. Anal. 260(9), 2742–2766 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lierl, J., Saloff-Coste, L.: Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms. arXiv:1205.6493 (2012)
  22. 22.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169(2), 903–991 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ma, Z.M., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)CrossRefGoogle Scholar
  24. 24.
    McCann, R.J., Topping, P.M.: Ricci flow, entropy and optimal transportation. Am. J. Math. 132(3), 711–730 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 (2002)
  26. 26.
    Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245 (2003)
  27. 27.
    Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math/0303109 (2003)
  28. 28.
    Savaré, G.: Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\(\infty \)) metric measure spaces. Discr. Cont. Dyn. Syst. A 34(4), 1641–1661 (2014)CrossRefGoogle Scholar
  29. 29.
    Sturm, K.-T.: On the geometry of metric measure spaces I and II. Acta Math. 169(1), 65–131 (2006)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sturm, K.-T.: Metric measure spaces with variable Ricci bounds and couplings of Brownian motions. In Festschrift Masatoshi Fukushima, volume 17 of Interdisciplinary Mathematical Sciences, pp. 553–575. World Science Publinsher, Hackensack (2015)Google Scholar
  31. 31.
    Sturm, K.-T.: Super Ricci flows for metric measure spaces, I. J. Funct. Anal. 275(12), 3504–3569 (2018)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Topping, P.: \(\cal{L}\)-optimal transportation for Ricci flow. J. Reine Angew. Math. 636, 93–122 (2009)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Villani, C.: Optimal Transport, Old and New. Springer, Berlin (2009)CrossRefGoogle Scholar
  34. 34.
    von Renesse, M.-K., Sturm, K.-T.: Transport inequalities, gradient estimates, entropy and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany

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