Journal of the Mathematical Society of Japan

Hypoelliptic Laplacian and probability

Jean-Michel BISMUT

Full-text: Open access

Abstract

The purpose of this paper is to describe the probabilistic aspects underlying the theory of the hypoelliptic Laplacian, as a deformation of the standard elliptic Laplacian. The corresponding diffusion on the total space of the tangent bundle of a Riemannian manifold is a geometric Langevin process, that interpolates between the geometric Brownian motion and the geodesic flow. Connections with the central limit theorem for the occupation measure by the geometric Brownian motion are emphasized. Spectral aspects of the hypoelliptic deformation are also provided on tori. The relevant hypoelliptic deformation of the Laplacian in the case of Riemann surfaces of constant negative curvature is briefly described, in connection with Selberg's trace formula.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1317-1357.

Dates
First available in Project Euclid: 27 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1445951153

Digital Object Identifier
doi:10.2969/jmsj/06741317

Mathematical Reviews number (MathSciNet)
MR3417500

Zentralblatt MATH identifier
1334.35019

Subjects
Primary: 35H10: Hypoelliptic equations 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

Keywords
hypoelliptic equations diffusion processes and stochastic analysis on manifolds

Citation

BISMUT, Jean-Michel. Hypoelliptic Laplacian and probability. J. Math. Soc. Japan 67 (2015), no. 4, 1317--1357. doi:10.2969/jmsj/06741317. https://projecteuclid.org/euclid.jmsj/1445951153


Export citation

References

  • M. F. Atiyah and I. M. Singer, The index of elliptic operators, I, Ann. of Math. (2), 87 (1968), 484–530.
  • M. F. Atiyah and I. M. Singer, The index of elliptic operators, III, Ann. of Math. (2), 87 (1968), 546–604.
  • J.-M. Bismut, The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc., 18 (2005), 379–476.
  • J.-M. Bismut, The hypoelliptic Dirac operator, In Geometry and dynamics of groups and spaces, Progr. Math., 265, Birkhäuser, Basel, 2008, 113–246.
  • J.-M. Bismut, Loop spaces and the hypoelliptic Laplacian, Comm. Pure Appl. Math., 61 (2008), 559–593.
  • J.-M. Bismut, A survey of the hypoelliptic Laplacian, Géométrie différentielle, physique mathématique, mathématiques et société, II, Astérisque, 322 (2008), 39–69.
  • J.-M. Bismut, Hypoelliptic Laplacian and orbital integrals, Annals of Mathematics Studies, 177, Princeton University Press, Princeton, NJ, 2011.
  • J.-M. Bismut, Index theory and the hypoelliptic Laplacian, In Metric and differential geometry, Progress in Mathematics, 297, Birkhäuser/Springer, Basel, 2012, 181–232.
  • J.-M. Bismut, Hypoelliptic Laplacian and Bott-Chern cohomology, Progress in Mathematics, 305, Birkhäuser/Springer, Cham, 2013, A theorem of Riemann-Roch-Grothendieck in complex geometry.
  • J.-M. Bismut and G. Lebeau, The hypoelliptic Laplacian and Ray-Singer metrics, Annals of Mathematics Studies, 167, Princeton University Press, Princeton, NJ, 2008.
  • J.-D. Deuschel and D. W. Stroock, Large deviations, Pure and Applied Mathematics, 137, Academic Press, Inc., Boston, MA, 1989.
  • J. Franchi and Y. Le Jan, Relativistic diffusions and Schwarzschild geometry, Comm. Pure Appl. Math., 60 (2007), 187–251.
  • J. Franchi and Y. Le Jan, Curvature diffusions in general relativity, Comm. Math. Phys., 307 (2011), 351–382.
  • J. Franchi and Y. Le Jan, Hyperbolic dynamics and Brownian motion, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2012. An introduction.
  • L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147–171.
  • A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2), 35 (1934), 116–117.
  • B. Kostant, Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the $\rho$-decomposition $C(\mathfrak g)={\rm End}\, V\sb \rho \otimes C(P)$, and the $\mathfrak g$-module structure of $\bigwedge \mathfrak g$, Adv. Math., 125 (1997), 275–350.
  • P. Langevin, Sur la théorie du mouvement brownien, C. R. Acad. Sci. Paris, 146 (1908), 530–533.
  • H. P. McKean, Selberg's trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math., 25 (1972), 225–246.
  • H. P. McKean, Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom., 1 (1967), 43–69.
  • D. Quillen, Superconnections and the Chern character, Topology, 24 (1985), 89–95.
  • D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math., 7 (1971), 145–210.
  • S. Shen, The hypoelliptic Laplacian, analytic torsion and Cheeger-Müller theorem, C. R. Math. Acad. Sci. Paris, 352 (2014), 153–156.
  • E. Witten, Supersymmetry and Morse theory, J. Differential Geom., 17 (1982), no.,4, 661–692 (1983).