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July 2004 Asymptotic behavior of divergences and Cameron–Martin theorem on loop spaces
Xiang Dong Li
Ann. Probab. 32(3B): 2409-2445 (July 2004). DOI: 10.1214/009117904000000045

Abstract

We first prove the Lp-convergence (p≥1) and a Fernique-type exponential integrability of divergence functionals for all Cameron–Martin vector fields with respect to the pinned Wiener measure on loop spaces over a compact Riemannian manifold. We then prove that the Driver flow is a smooth transform on path spaces in the sense of the Malliavin calculus and has an ∞-quasi-continuous modification which can be quasi-surely well defined on path spaces. This leads us to construct the Driver flow on loop spaces through the corresponding flow on path spaces. Combining these two results with the Cruzeiro lemma [J. Funct. Anal. 54 (1983) 206–227] we give an alternative proof of the quasi-invariance of the pinned Wiener measure under Driver’s flow on loop spaces which was established earlier by Driver [Trans. Amer. Math. Soc. 342 (1994) 375–394] and Enchev and Stroock [Adv. Math. 119 (1996) 127–154] by Doob’s h-processes approach together with the short time estimates of the gradient and the Hessian of the logarithmic heat kernel on compact Riemannian manifolds. We also establish the Lp-convergence (p≥1) and a Fernique-type exponential integrability theorem for the stochastic anti-development of pinned Brownian motions on compact Riemannian manifold with an explicit exponential exponent. Our results generalize and sharpen some earlier results due to Gross [J. Funct. Anal. 102 (1991) 268–313] and Hsu [Math. Ann. 309 (1997) 331–339]. Our method does not need any heat kernel estimate and is based on quasi-sure analysis and Sobolev estimates on path spaces.

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Xiang Dong Li. "Asymptotic behavior of divergences and Cameron–Martin theorem on loop spaces." Ann. Probab. 32 (3B) 2409 - 2445, July 2004. https://doi.org/10.1214/009117904000000045

Information

Published: July 2004
First available in Project Euclid: 6 August 2004

zbMATH: 1058.60039
MathSciNet: MR2078545
Digital Object Identifier: 10.1214/009117904000000045

Subjects:
Primary: 58G32 , 60H07

Keywords: divergence , Driver’s flow , exponential integrability , pinned Wiener measure , Quasi-invariance

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3B • July 2004
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