Journal of the Mathematical Society of Japan

A construction of diffusion processes associated with sub-Laplacian on CR manifolds and its applications

Hiroki KONDO and Setsuo TANIGUCHI

Full-text: Open access

Abstract

A diffusion process associated with the real sub-Laplacian $\Delta_b$, the real part of the complex Kohn–Spencer Laplacian $\square_b$, on a strictly pseudoconvex CR manifold is constructed via the Eells–Elworthy–Malliavin method by taking advantage of the metric connection due to Tanaka and Webster. Using the diffusion process and the Malliavin calculus, the heat kernel and the Dirichlet problem for $\Delta_b$ are studied in a probabilistic manner. Moreover, distributions of stochastic line integrals along the diffusion process will be investigated.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 1 (2017), 111-125.

Dates
First available in Project Euclid: 18 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1484730020

Digital Object Identifier
doi:10.2969/jmsj/06910111

Mathematical Reviews number (MathSciNet)
MR3597549

Zentralblatt MATH identifier
1362.32024

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

Keywords
CR manifold stochastic differential equation Malliavin calculus partial hypoellipticity Dirichlet problem

Citation

KONDO, Hiroki; TANIGUCHI, Setsuo. A construction of diffusion processes associated with sub-Laplacian on CR manifolds and its applications. J. Math. Soc. Japan 69 (2017), no. 1, 111--125. doi:10.2969/jmsj/06910111. https://projecteuclid.org/euclid.jmsj/1484730020


Export citation

References

  • S. Dragomir and G. Tomassini, Differential geometry and analysis on CR manifolds, Boston-Basel-Berlin, Birkhäuser, 2006.
  • K. D. Elworthy, Stochastic differential equations on manifolds, Cambridge University Press, 1982.
  • B. Gaveau, Principle de moindre action, propagation de la cheleur et estimees sous elliptiques sur certains groups nilpotents, Acta Math., 139 (1977), 95–153.
  • M. Gordina and T. Laetsch, A convergence to Brownian motion on sub-Riemannian manifolds, to appear in Trans. Amer. Math. Soc.
  • A. Greenleaf, The first eigenvalue of a sublaplacian on a pseudohermitian manifold, Comm. Partial Differential Equations., 10 (1985), 191–217.
  • E. P. Hsu, Stochastic analysis on manifolds, American Mathematical Soc., 2002.
  • N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland/Kodansha, Amsterdam/Tokyo, 1989.
  • H. Kunita, Supports of diffusion processes and controllability problems, In: Proc. Internat. Symp. on Stochastic differential equations (ed. K. Itô), Kinokuniya, Tokyo, 1978, 163–185.
  • J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc., 296 (1986), 411–429.
  • P. Malliavin, Géométrie différentielle stochastique, Univ. Montréal, 1978.
  • D. W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, American Mathematical Soc., 2005.
  • D. W. Stroock and S. Taniguchi, Regular points for the first boundary value problem associated with degenerate elliptic operators, in “Probability theory and harmonic anlaysis” (eds. J.-A. Chao and W.A. Woyczyński), Marcel Dekker, New York, 1986.
  • D. W. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math., 25 (1972), 651–713.
  • N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya Book Store Co., Ltd., Kyoto, 1975.
  • S. Taniguchi, Malliavin's stochastic calculus of variations for manifold-valued Wiener functionals and its applications, Z. Wahrsch. Verw. Gebiete., 65 (1983), 269–290.
  • S. Taniguchi, Applications of Malliavin's stochastic calculus to time-dependent systems of heat equations, Osaka J. Math., 22 (1985), 307–320.
  • S. M. Webster, Pseudo-hermitian structure on a real hypersurface, J. Differential Geom., 13 (1978), 25–41.