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

**Abstract**

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.

**English version** (Springer):
Ukrainian Mathematical Journal **56** (2004), no. 12, pp 1961-1974.

**Citation Example:** *Rudenko A. V.* Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup // Ukr. Mat. Zh. - 2004. - **56**, № 12. - pp. 1654-1664.

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