# 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.

Published

25.12.2004

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 56, no. 12, Dec. 2004, pp. 1654-6, https://umj.imath.kiev.ua/index.php/umj/article/view/3872.

Issue

Section

Research articles