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Approximation of densities of absolutely continuous parts of measures on a Hilbert space by the Ornstein-Uhlenbeck semigroup analysis. (Russian, English) Zbl 1103.28009
Ukr. Mat. Zh. 56, No. 12, 1654-1664 (2004); translation in Ukr. Math. J. 56, No. 12, 1961-1974 (2004).
Let $$\mu$$ be a Gaussian measure on a separable Hilbert space $$H$$, and let $$\nu$$ be another probability Borel measure on $$H$$. The Ornstein-Uhlenbeck semigroup $$T_t$$ acts on $$\nu$$ as follows: $(T_t\nu )(A)=\int\limits_H \int\limits_H \mathbf 1_A (e^{-t}x+\sqrt{1-e^{- 2t}}y)\nu (dx)\mu (dy).$ The author proves that the density of the part of $$T_t\nu$$, absolutely continuous with respect to $$\mu$$, converges in the measure $$\mu$$ to the density of $$\nu$$, as $$t\to 0$$. If $$\dim H<\infty$$, the convergence almost everywhere takes place; for a general $$H$$ that is true under some additional assumptions. Some conditions for the absolute continuity of $$T_t\nu$$ with respect to $$\mu$$ are also given.
##### MSC:
 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 60B11 Probability theory on linear topological spaces 47D07 Markov semigroups and applications to diffusion processes
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