Time-Localization of Random Distributions on Wiener Space II: Convergence, Fractional Brownian Density Processes

Abstract

For a random element X of a nuclear space of distributions on Wiener space C([0,1],Rd), the localization problem consists in “projecting” X at each time t∈[0,1] in order to define an S′(Rd)-valued process X={X(t),t∈[0,1]}, called the time-localization of X. The convergence problem consists in deriving weak convergence of time-localization processes (in C([0,1],S′(Rd)) in this paper) from weak convergence of the corresponding random distributions on C([0,1],Rd). Partial steps towards the solution of this problem were carried out in previous papers, the tightness having remained unsolved. In this paper we complete the solution of the convergence problem via an extension of the time-localization procedure. As an example, a fluctuation limit of a system of fractional Brownian motions yields a new class of S′(Rd)-valued Gaussian processes, the “fractional Brownian density processes”.