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