Nuclear Gelfand Triples on Wiener Space and Applications to Trajectorial Fluctuations of Particle Systems

Abstract

Motivated by the analysis of fluctuation limits of particle systems, a nuclear Gelfand triple is constructed on the space C([0,1], Rd) with σ-finite Wiener measure, following Kubo and Yokoi (1989, Nagoya Math. J.115, 139-149). The standard Gaussian measure on the space of distributions corresponds to "white noise" on C([0, 1], Rd). Using Lévy′s continuity theorem on nuclear spaces, this triple is applied to obtain trajectorial fluctuation limits of some particle systems. The limits are Gaussian random elements of the space of distributions on C([0, 1], Rd). The examples are a Poisson system of Brownian motions, a system of supercritical branching Brownian motions, and interacting diffusions with bounded and with linear interactions. In the latter examples the results of Sznitmann (1985, in "Infinite-dimensional Analysis and Stochastic Processes," Research Notes on Math., Vol. 124, pp. 145-160, Pitman, Boston, 1985) and of Tanaka and Hitsuda (1981, Hiroshima Math. J.11, 415-423) are sharpened. Previous trajectorial fluctuation limits for these models were known only in terms of characteristic functionals.