Asymptotic behavior of divergences and Cameron-Martin theorem on loop spaces

• 成果类型: Article
• 署名作者: Li, XD
• 署名单位: University of Oxford; Universite de Toulouse; Universite Toulouse III - Paul Sabatier
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000045
• 发表日期: 2004
• 页码: 2409-2445
• 关键词: compact riemannian manifold; quasi-invariance theorem; pinned brownian-motion; wiener space; differential-equations; Markov Process; sure analysis; heat kernel; path space; integration

摘要

We first prove the LP-convergence (p greater than or equal to 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 infinity-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 greater than or equal to 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.