Abstract
We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the Lp-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 Σ of zero Gaussian measure. To prove the equivalence we show the Wr,p(B,μ)-boundedness of certain smooth nonlinear truncation operators acting on potentials of nonnegative functions. We discuss connections to Gaussian Hausdorff measures. Roughly speaking, if Lp-uniqueness holds then the ‘removed’ set Σ must have sufficiently large codimension, in the case of the Ornstein-Uhlenbeck operator for instance at least 2p. For p = 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.
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Acknowledgments
We would like to thank Masanori Hino, Jun Masamune, Michael Röckner and Gerald Trutnau for inspiring and helpful discussions. We also thank the anonymous referee for valuable comments and for pointing out the time zero Gaussian free field as an important example, what inspired the writing of Sections ?? and ??.
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Research supported by the DFG IRTG 2235: ‘Searching for the regular in the irregular: Analysis of singular and random systems’.
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Hinz, M., Kang, S. Capacities, Removable Sets and Lp-Uniqueness on Wiener Spaces. Potential Anal 54, 503–533 (2021). https://doi.org/10.1007/s11118-020-09836-6
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DOI: https://doi.org/10.1007/s11118-020-09836-6
Keywords
- Wiener spaces
- Capacities
- Ornstein-Uhlenbeck operator
- Sobolev spaces
- Composition operators
- L p-uniqueness