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...
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| Datum: | 2004 |
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| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2004
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3872 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510007428120576 |
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| author | Rudenko, A. V. Руденко, А. В. Руденко, А. В. |
| author_facet | Rudenko, A. V. Руденко, А. В. Руденко, А. В. |
| author_sort | Rudenko, A. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:13:06Z |
| description | 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. |
| first_indexed | 2026-03-24T02:50:09Z |
| format | Article |
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| id | umjimathkievua-article-3872 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:50:09Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fb/e2ef804c438babd9b261386722bccffb.pdf |
| spelling | umjimathkievua-article-38722020-03-18T20:13:06Z Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup Приближение плотностей абсолютно непрерывных компонент мер и гильбертовом пространстве с помощью полугруппы Орнштейна — Уленбека Rudenko, A. V. Руденко, А. В. Руденко, А. В. 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. Вивчається поведінка мір, які є результатом дії пігрупи Орнштейна - Улеибека $T_t$, що пов'язана з гауссовою мірою $μ$, на довільну ймовірнісну міру $ν$ у сепарабельному гільбертовому просторі, при $t → 0+$. Доведено, що щільності абсолютно неперервних частин $T_tν$ по відношенню до $μ$ збігаються за мірою |і до щільності абсолютно неперервної частини V по підношенню до $μ$. У випадку скінченної вимірності простору доведено збіжність цих щільпостей $μ$-майже скрізь. У нескіпченновимірному випадку наведено деякі достатні умови для збіжності майже скрізь. Також розглянуто умови па абсолютну неперервність $T_tν$ по відношенню до $μ$. у термінах коефіцієнтів розкладу $T_tν$ в ряд за поліномами Ерміта (аналог розкладу Іто - Вінера) та зв'язок з фінітною абсолютною неперервністю. Institute of Mathematics, NAS of Ukraine 2004-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3872 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 12 (2004); 1654-1664 Український математичний журнал; Том 56 № 12 (2004); 1654-1664 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3872/4462 https://umj.imath.kiev.ua/index.php/umj/article/view/3872/4463 Copyright (c) 2004 Rudenko A. V. |
| spellingShingle | Rudenko, A. V. Руденко, А. В. Руденко, А. В. Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup |
| title | Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup |
| title_alt | Приближение плотностей абсолютно непрерывных компонент мер и гильбертовом пространстве с помощью полугруппы Орнштейна — Уленбека |
| title_full | Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup |
| title_fullStr | Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup |
| title_full_unstemmed | Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup |
| title_short | Approximation of Densities of Absolutely Continuous Components of Measures in a Hilbert Space Using the Ornstein-Uhlenbeck Semigroup |
| title_sort | approximation of densities of absolutely continuous components of measures in a hilbert space using the ornstein-uhlenbeck semigroup |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3872 |
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