We prove that Brownian motion on an abstract Wiener space B generates a locally equicontinuous semigroup on C b (B) equipped with the Tt-topology introduced by L. Le Cam. Hence we obtain a "Laplace operator" as its infinitesimal generator. Using this Laplacian, we discuss Poisson's equation and heat equation, and study its properties, especially the difference from the Gross Laplacian.
We show that Γ(Np), a subspace of exotic Hida-Kubo-Takenaka space is naturally imbedded into (E) * , the usual Hida-Kubo-Takenaka space under some conditions. We also study on Heat Equations associated with exotic Laplacians, such as Lévy Laplacian.
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