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Abstract

We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for theL(p)-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set sigma of zero Gaussian measure. To prove the equivalence we show theW(r,p)(B,mu)-boundedness of certain smooth nonlinear truncation operators acting on potentials of nonnegative functions. We discuss connections to Gaussian Hausdorff measures. Roughly speaking, ifL(p)-uniqueness holds then the 'removed' set sigma must have sufficiently large codimension, in the case of the Ornstein-Uhlenbeck operator for instance at least 2p. Forp= 2 we obtain parallel results on truncations, capacities and essential self-adjointness for Ornstein-Uhlenbeck operators with linear drift. These results apply to the time zero Gaussian free field as a prototype example.