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

Abstract

We first prove the L^p-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 L^p-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.