Diffusion by optimal transport in Heisenberg groups

Article

Abstract

We prove that the hypoelliptic diffusion of the Heisenberg group \({\mathbb{H }}_n\) describes, in the space of probability measures over \({\mathbb{H }}_n\), a curve driven by the gradient flow of the Boltzmann entropy \({{\mathrm{Ent}}}\), in the sense of optimal transport. We prove that conversely any gradient flow curve of \({{\mathrm{Ent}}}\) satisfy the hypoelliptic heat equation. This occurs in the subRiemannian \({\mathbb{H }}_n\), which is not a space with a lower Ricci curvature bound in the metric sense of Lott–Villani and Sturm.

Keywords

Heisenberg group Optimal transport theory Gradient flow 

Mathematics Subject Classification (2000)

28A33 53C17 60J60 

Notes

Acknowledgments

I wish to thank Hervé Pajot and Karl-Theodor Sturm that were my advisers when I started this research. I thank Luigi Ambrosio, Nicola Gigli and Cédric Villani for helpful discussions as well as my ex-colleagues Matthias Erbar and Miguel Rodrigues. The referee of the first version gave me good advice (for instance on alternative proofs in Proposition 3.1 and Paragraph 4.1) and the motivation for avoiding a non-necessary extra hypothesis in Paragraph 4.2. I would like to thank her or him very much. I thank also the other referees for their careful reading.

References

  1. 1.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)Google Scholar
  2. 2.
    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 Verlag, Basel (2008)Google Scholar
  3. 3.
    Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Preprint (2011)Google Scholar
  4. 4.
    Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Preprint (2011)Google Scholar
  5. 5.
    Ambrosio, L., Rigot, S.: Optimal mass transportation in the Heisenberg group. J. Funct. Anal. 208(2), 261–301 (2004)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Ambrosio, L., Savaré, G.: Gradient flows of probability measures. In: Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., pp. 1–136. Elsevier/North-Holland, Amsterdam (2007)Google Scholar
  7. 7.
    Bernard, P.: Young measures, superposition and transport. Indiana Univ. Math. J. 57(1), 247–275 (2008)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Blachman, N.M.: The Convolution Inequality for Entropy Powers. IEEE Trans. Inform. Theory IT 11(2), 267–271 (1965)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Capogna, L., Danielli, D., Pauls, S.D., Tyson, J.T.: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, vol. 259 of Progress in Mathematics. Birkhäuser Verlag, Basel (2007)Google Scholar
  10. 10.
    Cheeger, J., Kleiner, B.: Differentiating maps into \(L^1\), and the geometry of BV functions. Ann. Math. (2) 171(2), 1347–1385 (2010)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    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)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Fang, S., Shao, J., Sturm, K.-T.: Wasserstein space over the Wiener space. Probab. Theory Relat. Fields 146, 535–565 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Fathi, A., Figalli, A.: Optimal transportation on non-compact manifolds. Israel J. Math. 175, 1–59 (2010)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Figalli, A., Juillet, N.: Absolute continuity of Wasserstein geodesics in the Heisenberg group. J. Funct. Anal. 255(1), 133–141 (2008)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Figalli, A., Rifford, L.: Mass transportation on sub-Riemannian manifolds. Geom. Funct. Anal. 20(1), 124–159 (2010)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups, Vol. 28 of Mathematical Notes. Princeton University Press, Princeton (1982)Google Scholar
  17. 17.
    Franchi, B., Serapioni, R., Serra Cassano, F.: Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321(3), 479–531 (2001)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math. 139(1–2), 95–153 (1977)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    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)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Gigli, N., Kuwada, K., Ohta, S.-i.: Heat flow on Alexandrov spaces. Commun. Pure Appl. Math. 3; 307–331 (2013)Google Scholar
  21. 21.
    Gigli, N., Ohta, S.-I.: First variation formula in Wasserstein spaces over compact Alexandrov spaces. Canad. Math. Bull. 55(4), 723–735 (2012)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Heinonen, J.: Calculus on Carnot groups. In: Fall School in Analysis (Jyväskylä 1994), vol. 68 of report, pp. 1–31. University of Jyväskylä, Jyväskylä (1995)Google Scholar
  23. 23.
    Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)MATHMathSciNetGoogle Scholar
  25. 25.
    Juillet, N.: Optimal transport and geometric analysis in Heisenberg groups. PhD thesis, Université Grenoble 1, Bonn Universität, http://tel.archives-ouvertes.fr/tel-00345301 (2008)
  26. 26.
    Juillet, N.: Geometric inequalities and generalized Ricci bounds in the Heisenberg group. Int. Math. Res. Not. (2009)Google Scholar
  27. 27.
    Khesin, B., Lee, P.: A nonholonomic Moser theorem and optimal transport. J. Symplectic Geom. 7(4), 381–414 (2009)MATHMathSciNetGoogle Scholar
  28. 28.
    Korányi, A., Reimann, H.M.: Quasiconformal mappings on the Heisenberg group. Invent. Math. 80(2), 309–338 (1985)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Li, H.-Q.: Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg. J. Funct. Anal. 236(2), 369–394 (2006)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Li, H.-Q.: Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg. C. R. Math. Acad. Sci. Paris 344(8), 497–502 (2007)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Lott, J., Villani, C.: Weak curvature conditions and functional inequalities. J. Funct. Anal. 245(1), 311–333 (2007)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Maas, J.: Gradient flows of the entropy for finite markov chains. J. Funct. Anal. 261(8), 2250–2292 (2011)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    McCann, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589–608 (2001)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications, Vol. 91 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2002)Google Scholar
  36. 36.
    Ohta, S-i: Gradient flows on Wasserstein spaces over compact Alexandrov spaces. Am. J. Math. 131(2), 475–516 (2009)CrossRefMATHGoogle Scholar
  37. 37.
    Ohta, S-i, Sturm, K.-T.: Heat flow on Finsler manifolds. Commun. Pure Appl. Math. 62(10), 1386–1433 (2009)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    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)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Petrunin, A.: Alexandrov meets Lott–Villani–Sturm. Münster J. Math. 4, 53–64 (2011)MATHMathSciNetGoogle Scholar
  41. 41.
    Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(3–4), 247–320 (1976)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Savaré, G.: Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds. C. R. Math. Acad. Sci. Paris 345(3), 151–154 (2007)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Stam, A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control 2, 101–112 (1959)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Vol. 43 of Princeton Mathematical Series. Princeton University Press, Princeton (with the assistance of Timothy S. Murphy). Monographs in Harmonic Analysis III (1993)Google Scholar
  45. 45.
    Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Villani, C.: Optimal Transport, Vol. 338 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin-Heidelberg (2009)Google Scholar
  48. 48.
    Zhang, H.-C., Zhu, X.-P.: Ricci curvature on Alexandrov spaces and rigidity theorems. Commun. Anal. Geom. 18(3), 503–553 (2010)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique Avancée, UMR 7501Université de Strasbourg et CNRSStrasbourgFrance

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