Electronic Communications in Probability

Martingale Representation and a Simple Proof of Logarithmic Sobolev Inequalities on Path Spaces

Mireille Capitaine, Elton Hsu, and Michel Ledoux

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We show how the Clark-Ocone-Haussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the Ornstein-Uhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space. By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diffusion measure as long as the torsion of the corresponding Riemannian connection satisfies Driver's total antisymmetry condition.

Article information

Electron. Commun. Probab., Volume 2 (1997), paper no. 7, 71-81.

Accepted: 15 December 1997
First available in Project Euclid: 26 January 2016

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Zentralblatt MATH identifier

Primary: 58G32

Martingale representation logarithmic Sobolev inequality Brownian motion Riemannian manifold

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Capitaine, Mireille; Hsu, Elton; Ledoux, Michel. Martingale Representation and a Simple Proof of Logarithmic Sobolev Inequalities on Path Spaces. Electron. Commun. Probab. 2 (1997), paper no. 7, 71--81. doi:10.1214/ECP.v2-986. https://projecteuclid.org/euclid.ecp/1453832502

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