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Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis 1 (1967), 123-181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator N is defined by Nf(x) = -lim∈←0{E[f(U(τ∈ξ))] - f(x)}/E[τ∈ξ, where τxε{lunate} is the first exit time of U(t) starting at x from the ball of radius ε{lunate} with center x. It is shown that Nf(x) = -trace D2f(x)+〈Df(x),x〉 for a large class of functions f. Let rt(x, dy) be the transition probabilities of U(t). The λ-potential Gλf, λ > 0, and normalized potential Rf of f are defined by Gλf(X) = ∫0∞ e-λtrtf(x) dt and Rf(x) = ∫0∞ [rtf(x) - rtf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D2Gλf(x) - 〈DGλf(x), x〉 = -f(x) + λGλf(x) and trace D2Rf(x) - 〈DRf(x), x〉 = -f(x) + ∫Bf(y)p1(dy), where p1 is the Wiener measure in B with parameter 1. Some approximation theorems are also proved. © 1976.

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Journal of Functional Analysis

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