Time-Localization of Random Distributions on Wiener Space

A nuclear space of distributions on Wiener space was constructed by Gorostiza and Nualart 10] as a framework for studying weak convergence of trajectorial fluctuations of particle systems. A basic problem in recovering the usual time-evolution results from the trajectorial ones consists in associating in a unique way an -valued process to a random distribution on by localizing it at each time t 0,1]. In this paper we solve this problem for a large class of random distributions which includes trajectorial fluctuation limits of some systems of diffusions.