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Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup

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We study the behavior of measures obtained as a result of the action of the Ornstein-Uhlenbeck semigroup T t associated with the Gaussian measure μ on an arbitrary probability measure ν in a separable Hilbert space as t → 0+. We prove that the densities of the parts of T t ν absolutely continuous with respect to μ converge in the measure μ to the density of the part of ν absolutely continuous with respect to μ. For a finite-dimensional space, we prove the convergence of these densities μ-almost everywhere. In the infinite-dimensional case, we give sufficient conditions for almost-everywhere convergence. We also consider conditions on the absolute continuity of T t ν with respect to μ in terms of the coefficients of the expansion of T t ν in a series in Hermite polynomials (an analog of the Ito- Wiener expansion) and the connection with finite absolute continuity.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1654 – 1664, December, 2004.

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Rudenko, A.V. Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup. Ukr Math J 56, 1961–1974 (2004). https://doi.org/10.1007/s11253-005-0161-3

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  • DOI: https://doi.org/10.1007/s11253-005-0161-3

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