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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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