Skip to main content
Log in

Super-Ricci flows and improved gradient and transport estimates

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

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.)

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  3. Ambrosio, L., Gigli, N., Savaré, G.: Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43(1), 339–404 (2015)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Book  Google Scholar 

  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)

  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 Statistics

  8. Bauer, H.: Wahrscheinlichkeitstheorie. de Gruyter Lehrbuch. [de Gruyter Textbook]. Walter de Gruyter & Co., Berlin, 5th edn (2002)

  9. Bogachev, V.: Measure Theory, vol. 1. Springer, Berlin (2007)

    Book  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  11. Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  14. Nicola Gigli. Nonsmooth differential geometry-An approach tailored for spaces with Ricci curvature bounded from below. arXiv:1407.0809 (2014)

  15. Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Am. Math. Soc. 236(1113), vi–91 (2015)

    Article  MathSciNet  Google Scholar 

  16. Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)

    Article  MathSciNet  Google Scholar 

  17. Haslhofer, R., Naber, A.: Weak solutions for the Ricci flow I. arXiv:1504.00911 (2015)

  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. Kuwada, K.: Duality on gradient estimates and Wasserstein controls. J. Funct. Anal. 258(11), 3758–3774 (2010)

    Article  MathSciNet  Google Scholar 

  20. Kuwada, K., Philipowski, R.: Coupling of Brownian motions and Perelman’s L-functional. J. Funct. Anal. 260(9), 2742–2766 (2011)

    Article  MathSciNet  Google Scholar 

  21. Lierl, J., Saloff-Coste, L.: Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms. arXiv:1205.6493 (2012)

  22. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169(2), 903–991 (2009)

    Article  MathSciNet  Google Scholar 

  23. Ma, Z.M., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)

    Book  Google Scholar 

  24. McCann, R.J., Topping, P.M.: Ricci flow, entropy and optimal transportation. Am. J. Math. 132(3), 711–730 (2010)

    Article  MathSciNet  Google Scholar 

  25. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 (2002)

  26. Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245 (2003)

  27. Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math/0303109 (2003)

  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)

    Article  Google Scholar 

  29. Sturm, K.-T.: On the geometry of metric measure spaces I and II. Acta Math. 169(1), 65–131 (2006)

    Article  MathSciNet  Google Scholar 

  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. Sturm, K.-T.: Super Ricci flows for metric measure spaces, I. J. Funct. Anal. 275(12), 3504–3569 (2018)

    Article  MathSciNet  Google Scholar 

  32. Topping, P.: \(\cal{L}\)-optimal transportation for Ricci flow. J. Reine Angew. Math. 636, 93–122 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Villani, C.: Optimal Transport, Old and New. Springer, Berlin (2009)

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Eva Kopfer.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kopfer, E. Super-Ricci flows and improved gradient and transport estimates. Probab. Theory Relat. Fields 175, 897–936 (2019). https://doi.org/10.1007/s00440-019-00904-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-019-00904-6

Mathematics Subject Classification

Navigation