Electronic Journal of Probability

The Abstract Riemannian Path Space

D. Feyel and A. de La Pradelle

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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

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