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We prove that Brownian motion on an abstract Wiener space B generates a locally equicontinuous semigroup on $C_b\left(B\right)$ equipped with the $T_t$-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.