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The hypoelliptic Laplacian of J.-M. Bismut

  • Gilles Lebeau
Chapter

Abstract

In recents works, J.-M. Bismut has introduced an “hypoelliptic Laplacian” acting on differentials forms on the cotangent bundle T*X of a Riemannian compact manifold X. This operator is a deformation of the Hodge Laplacian on X. We present here some analytic properties of this new operator.

Keywords

Bilinear Form Heat Kernel Cotangent Bundle Riemannian Compact Manifold Degenerate Minimum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Bis04a]
    J.-M. Bismut. Le Laplacien hypoelliptique. In Séminaire: Équations aux Dérivées Partielles, 2003–2004, Sémin. Équ. Dériv. Partielles, pages Exp. No. XXII, 15. École Polytech., Palaiseau, 2004.Google Scholar
  2. [Bis04b]
    J.-M. Bismut. Le Laplacien hypoelliptique sur le fibré cotangent. C. R. Math. Acad. Sci. Paris Sér. I, 338:555–559, 2004.MATHMathSciNetGoogle Scholar
  3. [Bis04c]
    J.-M. Bismut. Une déformation de la théorie de Hodge sur le fibré cotangent. C. R. Acad. Sci. Paris Sér. I, 338:471–476, 2004.MATHMathSciNetGoogle Scholar
  4. [Bis05]
    J.-M. Bismut. The hypoelliptic Laplacian on the cotangent bundle. To appear in J.A.M.S., 2005.Google Scholar
  5. [BL06]
    J.-M. Bismut and G. Lebeau. The hypoelliptic Laplacian and Ray-Singer metrics. to appear, 2006.Google Scholar
  6. [DV01]
    L. Desvillettes and C. Villani. On the trend to equilibrium in spatially inhomogeneous entropy dissipating systems: the linear Fokker Planck equation. CPAM, 54:1–42, 2001.MATHMathSciNetGoogle Scholar
  7. [HN04]
    F. Hérau and F. Nier. Isotropic hypoellipticity and trend to equilibrium for Fokker-Planck equations with high degree potential. Arch. Ration. Mecha. Anal., 171(2):151–218, 2004.MATHCrossRefGoogle Scholar
  8. [HN05]
    B. Helffer and F. Nier. Hypoellipticity and spectral theory for Fokker-Planck operators and Witten Laplacians. Lect. Notes in Math., 1862, 2005.Google Scholar
  9. [HSS05]
    F. Hérau, J. Sjostrand, and C. Stolk. Semiclasical analysis for the Kramers-Fokker-Planck equation. CPDE, 30:689–760, 2005.MATHCrossRefGoogle Scholar
  10. [Leb05]
    G. Lebeau. Geometric Fokker Planck equations. Portugaliae Mathematica, 62(4), 2005.Google Scholar
  11. [Leb06]
    G. Lebeau. Equations de Fokker Planck géométriques II: estimations hypoelliptiques maximales. to appear, 2006.Google Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  • Gilles Lebeau
    • 1
  1. 1.Département de MathématiquesUniversité de Nice Sophia-Antipolis, Parc ValroseNice Cedex 02France

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