Siochastic Analysis on loop spaces

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The results obtained for 1998-1999 are as follows :1. We proved a Varadhan type asymptotics of diffusion processes in Wiener spaces2. We gave an irreducibility criterion of diffusion processes on a subset in an Wiener space in terms of a certain connectivity of the subset.3. We proved a Clark-Ocone formula for pinned Brownian motion.The following are explanations of the aboves.1. : This is a generalization of the Fang's results for Ornstein-Uhlenbeck processes to more general diffusion coefficients.2. : The head investigator proved an irreducibility theorem for diffusion processes on loop spaces using Kusuoka's theorem. In this work, the head investigator proved the generalization.3. : As an application, the head investigator proved a logarithmic Sobolev inequality on loop space over hyperbolic spaces.