Electronic Journal of Probability

The Abstract Riemannian Path Space

D. Feyel and A. de La Pradelle

Full-text: Open access

Abstract

On the Wiener space $\Omega$, we introduce an abstract Ricci process $A_t$ and a pseudo-gradient $F\rightarrow{F}^\sharp$ which are compatible through an integration by parts formula. They give rise to a $\sharp$-Sobolev space on $\Omega$, logarithmic Sobolev inequalities, and capacities, which are tight on Hoelder compact sets of $\Omega$. These are then applied to the path space over a Riemannian manifold.

Article information

Source
Electron. J. Probab., Volume 5 (2000), paper no. 11, 17 pp.

Dates
Accepted: 25 May 2000
First available in Project Euclid: 7 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1457376446

Digital Object Identifier
doi:10.1214/EJP.v5-67

Mathematical Reviews number (MathSciNet)
MR1781023

Zentralblatt MATH identifier
0949.60064

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H25: Random operators and equations [See also 47B80] 58D20: Measures (Gaussian, cylindrical, etc.) on manifolds of maps [See also 28Cxx, 46T12] 58J99: None of the above, but in this section

Keywords
Wiener space Sobolev spaces Bismut-Driver formula Logarithmic Sobolev inequality Capacities Riemannian manifold path space

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Feyel, D.; de La Pradelle, A. The Abstract Riemannian Path Space. Electron. J. Probab. 5 (2000), paper no. 11, 17 pp. doi:10.1214/EJP.v5-67. https://projecteuclid.org/euclid.ejp/1457376446


Export citation

References

  • S. Aida, K. D. Elworthy, Differential calculus on path and loop spaces I. Logarithmic Sobolev inequalities on path spaces. C.R.A.S. Paris 321, (1995), 97-102
  • J.-M. Bismut, Large Deviations and the Malliavin Calculus. Birkhäuser (1984)
  • M. Capitaine, E. P. Hsu, M. Ledoux, Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Elect. Comm. in Prob. 2, (1997), 71-81
  • A. B. Cruzeiro, P. Malliavin, Renormalized differential geometry on path space: structural equation, curvature. J. Funct. Anal. 140, (1996), 381-448.
  • J. Deny, Méthodes hilbertiennes en théorie du potentiel. CIME. Potential theory (1970)
  • J. Deny, J.-L. Lions, Espaces du type de Beppo-Levi. Ann. I. Fourier, III, (1953), 305-370.
  • B. K. Driver, A Cameron-Martin type quasi-invariance theorem for Brownian motion compact Riemannian manifold. J. Funct. Anal. 110, (1992), 272-376.
  • B. K. Driver, The non-equivalence of the Dirichlet form on path spaces. Proc. U.S.-Japan bilateral seminar (1994)
  • B. K. Driver, Integration by parts for heat kernel measure revisited. J. Math. Pures Appl. 76:9, 703-737 (1997)
  • B. K. Driver, M. Röckner, Construction of diffusions on path and loop spaces of compact Riemannian manifolds. C.R.A.S. Paris, 315, (1992) 603-608
  • K. D. Elworthy, X. M. Li, A class of integration by parts formulae in stochastic analysis I. In Itô Stochastic Calculus and probability theory, 15-30. N. Ikeda and al., Springer Verlag, Tokyo (1996).
  • K. D. Elworthy, Y. Le Jan, X. M. Li, Integration by parts formula for degenerate diffusion measures on path spaces. C.R.A.S. Paris, 323, (1996), 921-926
  • O. Enchev, D. Stroock, Towards a Riemannian geometry on the path space over a Riemannian manifold. J. Funct. Anal. 134, (1995) 392-416
  • S. Fang, Inégalité du type de Poincaré sur l'espace des chemins riemanniens. C.R.A.S. Paris, 318, (1994), 257-260
  • S. Fang, P. Malliavin, Stochastic Analysis on the Path Space of a Riemannian manifold. J. Funct. Anal. 118, (1993), 249-274
  • D. Feyel, Transformations de Hilbert-Riesz. CRAS, Paris, 310, (1990), 653-655
  • D. Feyel, A. de La Pradelle, Représentation d'espaces de Riesz-Banach sur des espaces quasi-topologiques. Bull. Acad. Royale de Belgique, 5eme série, t. LXIV, (1978-79), 340-350
  • D. Feyel, A. de La Pradelle, Espaces de Sobolev gaussiens. Ann. I. Fourier, 39:4 (1989) 875-908
  • D. Feyel, A. de La Pradelle, Capacités gaussiennes. Ann. I. Fourier, 41:1 (1991) 49-76
  • D. Feyel, A. de La Pradelle, Brownian Processes in Infinite Dimension. Potential Analysis, 4, (1995), p.173-183
  • D. Feyel, A. de La Pradelle, On the approximate solutions of the Stratonovitch equation. Elect. J. Prob., 3:7, (1998), 1-14
  • D. Feyel, A. de La Pradelle, Fractional integrals and Brownian processes. Potential Analysis, 10, (1999), p.273-288
  • L. Gross, Logarithmic Sobolev inequalities. Amer. J. Math. 97, (1975) 1061-1083
  • L. Gross, Logarithmic Sobolev inequalities for the heat kernel of a Lie group. In White Noise Analysis (T. Hida and al., Eds), p. 108-130, World Scientific, Singapore-Teaneck-New Jersey (1990)
  • E. P. Hsu, Inégalités de Sobolev logarithmiques sur un espace de chemins. C.R.A.S. Paris, 320, (1995) 1009-1012
  • N. Ikeda, S. Watanabe, Stochastic Differential Equation and Diffusion Processes. North-Holland, Amsterdam-Oxford-New York, (1981)
  • P. Malliavin, Stochastic Analysis. Springer Verlag (1997)
  • D. Nualart, The Malliavin calculus and Related Topics. Springer Verlag (1995)
  • I. Shigekawa, A quasihomeomorphism of the Wiener space. Proc. Sump. Pure and Appllied Mathematics 57 (Cranston and Pinsky Ed.) Cornell, (1993) Am. Math. Soc. 8, (1995), 473-487
  • F. Y. Wang, Logarithmic Sobolev inequalities for diffusion processes with application to path space. Chinese J. Appl. Probab. Stat. 12:3, (1996) 255-264