Advertisement

Curvature bounds for configuration spaces

  • Matthias Erbar
  • Martin Huesmann
Article
  • 180 Downloads

Abstract

We show that the configuration space \(\Upsilon \) over a manifold \(M\) inherits many curvature properties of the manifold. For instance, we show that a lower Ricci curvature bound on \(M\) implies a lower Ricci curvature bound on \(\Upsilon \) in the sense of Lott–Sturm–Villani, the Bochner inequality, gradient estimates and Wasserstein contraction. Moreover, we show that the heat flow on \(\Upsilon \) can be identified as the gradient flow of the entropy.

Notes

Acknowledgments

The authors would like to thank Theo Sturm and Fabio Cavalletti for several fruitful discussions on the subject of this paper.

References

  1. 1.
    Albeverio, S., Kondratiev, Y.G., Röckner, M.: Analysis and geometry on configuration spaces. J. Funct. Anal. 154(2), 444–500 (1998)Google Scholar
  2. 2.
    Albeverio, S., Kondratiev, Y.G., Röckner, M.: Analysis and geometry on configuration spaces: the Gibbsian case. J. Funct. Anal. 157(1), 242–291 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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 (2008)zbMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Preprint at arXiv:1109.0222 (2011)
  5. 5.
    Ambrosio, L., Gigli, N., Savaré, G.: Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Preprint at arXiv:1209.5786 (2012)
  6. 6.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bakry, D., Émery, M.: Diffusions hypercontractives. In Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math., pp. 177–206. Springer, Berlin (1985)Google Scholar
  8. 8.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, volume 33. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  9. 9.
    Chodosh, O.: A lack of Ricci bounds for the entropic measure on Wasserstein space over the interval. J. Funct. Anal. 262(10), 4570–4581 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Daneri, S., Savaré, G.: Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Anal. 40(3), 1104–1122 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Decreusefond, L.: Wasserstein distance on configuration space. Potential Anal. 28(3), 283–300 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Deng, C.-S.: Harnack inequality on configuration spaces: the coupling approach and a unified treatment. Stoch. Process. Appl. 124(1), 220–234 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Erbar, M., Kuwada, K., Sturm, K.-T.: On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. arXiv preprint arXiv:1303.4382 (2013)
  15. 15.
    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
  16. 16.
    Kallenberg, O.: Foundations of Modern Probability. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kellerer, H.G.: Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete 67(4), 399–432 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kendall, D.G.: On infinite doubly-stochastic matrices and Birkhoff’s problem 111. J. Lond. Math. Soc. 35, 81–84 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kondratiev, Y.G., Lytvynov, E., Röckner, M.: Non-equilibrium stochastic dynamics in continuum: the free case. Condens. Matter Phys. 11(4), 701–721 (2008)CrossRefGoogle Scholar
  20. 20.
    LaFontaine, J., Katz, M., Gromov, M., Bates, S.M., Pansu, P., Semmes, S.: Metric Structures for Riemannian and Non-Riemannian Spaces. Springer, Berlin (2007)Google Scholar
  21. 21.
    Lebedeva, N., Petrunin, A.: Curvature bounded below: a definition a la Berg-Nikolaev. Electron. Res. Announc. Math. Sci. 17, 122–124 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lisini, S.: Absolutely continuous curves in extended Wasserstein-Orlicz spaces. arXiv preprint arXiv:1402.7328 (2014)
  23. 23.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 163(3), 903–991 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Naber, A.: Characterizations of bounded Ricci curvature on smooth and nonsmooth spaces. arXiv preprint arXiv:1306.6512 (2013)
  25. 25.
    Osada, H.: Infinite-dimensional stochastic differential equations related to random matrices. Probab. Theory Relat. Fields 153(3–4), 471–509 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Osada, H.: Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials. Ann. Probab. 41(1), 1–49 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Otto, F., Westdickenberg, M.: Eulerian calculus for the contraction in the Wasserstein distance. SIAM J. Math. Anal. 37(4), 1227–1255 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Privault, N.: Connections and curvature in the Riemannian geometry of configuration spaces. J. Funct. Anal. 185(2), 367–403 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Röckner, M., Schied, A.: Rademacher’s theorem on configuration spaces and applications. J. Funct. Anal. 169(2), 325–356 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Srivastava, S.M.: A Course on Borel Sets, Volume 180. Springer, Berlin (1998)CrossRefGoogle Scholar
  31. 31.
    Stroock, D.W.: An Introduction to the Analysis of Paths on a Riemannian Manifold, Volume 74 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2000)Google Scholar
  32. 32.
    Sturm, K.-T.: Metric spaces of lower bounded curvature. Expo. Math. 17(1), 35–47 (1999)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Sturm, K.T.: On the geometry of metric measure spaces.I. Acta Math. 196(1), 65–131 (2006)MathSciNetCrossRefGoogle Scholar
  34. 34.
    von Renesse, M.-K., Sturm, K.-T.: Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl. Math. 58(7), 923–940 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute for Applied MathematicsUniversity of BonnBonnGermany

Personalised recommendations