Skip to main content
Log in

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

  • Published:
Inventiones mathematicae Aims and scope

Abstract

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces \((X,\mathsf {d},\mathfrak {m})\). Our main results are:

  • A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d).

  • The equivalence of the heat flow in \(L^{2}(X,\mathfrak {m})\) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional \(\mathrm {Ent}_{\mathfrak {m}}\) in the space of probability measures .

  • The proof of density in energy of Lipschitz functions in the Sobolev space \(W^{1,2}(X,\mathsf {d},\mathfrak {m})\).

  • A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper.

Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169:903–991, 2009) and Sturm (Acta Math. 196: 65–131, 2006, and Acta Math. 196:133–177, 2006) and require neither the doubling property nor the validity of the local Poincaré inequality.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agueh, M.: Existence of solutions to degenerate parabolic equations via the Monge–Kantorovich theory. Adv. Differ. Equ. 10, 309–360 (2005)

    MATH  MathSciNet  Google Scholar 

  2. Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Piccoli, B., Rascle, M. (eds.) Modelling and Optimisation of Flows on Networks, Cetraro, Italy, 2009. Lecture Notes in Mathematics, pp. 1–155. Springer, Berlin (2013)

    Chapter  Google Scholar 

  3. Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. arXiv:1109.0222v1

  4. Ambrosio, L., Gigli, N., Savaré, G.: Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. arXiv:1111.3730 (2011)

  5. 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, Basel (2008)

    MATH  Google Scholar 

  6. Ambrosio, L., Savaré, G., Zambotti, L.: Existence and stability for Fokker–Planck equations with log-concave reference measure. Probab. Theory Relat. Fields 145, 517–564 (2009)

    Article  MATH  Google Scholar 

  7. Ambrosio, L., DiMarino, S.: Equivalent definitions of BV space and of total variation on metric measure spaces (preprint, 2012)

  8. Bogachev, V.I.: Gaussian Measures. Mathematical Surveys and Monographs, vol. 62. Am. Math. Soc., Providence (1998)

    Book  MATH  Google Scholar 

  9. Bogachev, V.I.: Measure Theory, vols. I, II. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  10. Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, vol. 5. North-Holland, Amsterdam (1973). Notas de Matemática (50)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Mathematics Studies, vol. 29. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  13. Erbar, M.: The heat equation on manifolds as a gradient flow in the Wasserstein space. Ann. Inst. Henri Poincaré Probab. Stat. 46, 1–23 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fang, S., Shao, J., Sturm, K.-T.: Wasserstein space over the Wiener space. Probab. Theory Relat. Fields 146, 535–565 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Feyel, D., Üstünel, A.S.: Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theory Relat. Fields 128, 347–385 (2004)

    Article  MATH  Google Scholar 

  16. Gigli, N.: On the heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. Partial Differ. Equ. 39(1–2), 101–120 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gigli, N., Kuwada, K., Ohta, S.: Heat flow on Alexandrov spaces (submitted, 2010)

  18. Gigli, N.: On the differential structure of metric measure spaces and applications. arXiv:1205.6622

  19. Gozlan, N., Roberto, C., Samson, P.: Hamilton–Jacobi equations on metric spaces and transport entropy inequalities (preprint, 2011)

  20. Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hajłasz, P., Koskela, P.: Sobolev Met Poincaré. Mem. Amer. Math. Soc., vol. 145 (2000), pp. x+101

    Google Scholar 

  22. Heinonen, J.: Nonsmooth calculus. Bull. Am. Math. Soc. 44, 163–232 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lisini, S.: Characterization of absolutely continuous curves in Wasserstein spaces. Calc. Var. Partial Differ. Equ. 28, 85–120 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lisini, S.: paper in preparation

  27. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. ArXiv Mathematics e-prints (2004)

  28. Lott, J., Villani, C.: Hamilton-Jacobi semigroup on length spaces and applications. J. Math. Pures Appl. 88, 219–229 (2007)

    MATH  MathSciNet  Google Scholar 

  29. Lott, J., Villani, C.: Weak curvature conditions and functional inequalities. J. Funct. Anal. 245, 311–333 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Ohta, S.-I.: Finsler interpolation inequalities. Calc. Var. Partial Differ. Equ. 36, 211–249 (2009)

    Article  MATH  Google Scholar 

  32. Ohta, S.-I., Sturm, K.-T.: Heat flow on Finsler manifolds. Commun. Pure Appl. Math. 62, 1386–1433 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  33. Ohta, S.-I., Sturm, K.-T.: Non-contraction of heat flow on Minkowski spaces (preprint, 2010)

  34. Otto, F.: Doubly degenerate diffusion equations as steepest descent (manuscript, 1996)

  35. Rajala, T.: Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm. J. Funct. Anal. 263, 896–924 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Rajala, T.: Local Poincaré inequalities from stable curvature conditions on metric spaces. Calc. Var. Partial Differ. Equ. 44, 477–494 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. Schwartz, L.: Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Tata Institute of Fundamental Research Studies in Mathematics, vol. 6. Bombay by Oxford University Press, London (1973). Published for the Tata Institute of Fundamental Research

    MATH  Google Scholar 

  38. Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16, 243–279 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  39. Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196, 65–131 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  40. Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196, 133–177 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  41. Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the support of the ERC ADG GeMeThNES and of the PRIN08-grant from MIUR for the project Optimal transport theory, geometric and functional inequalities, and applications. The authors warmly thank an anonymous reviewer for his extremely detailed and constructive report.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Luigi Ambrosio.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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, 289–391 (2014). https://doi.org/10.1007/s00222-013-0456-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-013-0456-1

Mathematics Subject Classification

Navigation