On generalized local time for the process of brownian motion
We prove that the functionals \(\delta _\Gamma (B_t ) and \frac{{\partial ^k }}{{\partial x_1^k ...\partial x_d^{k_d } }}\delta _\Gamma (B_t ), k_1 + ... + k_d = k > 1,\) of a d-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δΓ(·) is a gen...
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| Date: | 2000 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2000
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4405 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510534042910720 |
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| author | Вакип, V. V. Бакун, В. В. Бакун, В. В. |
| author_facet | Вакип, V. V. Бакун, В. В. Бакун, В. В. |
| author_sort | Вакип, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:28:32Z |
| description | We prove that the functionals \(\delta _\Gamma (B_t ) and \frac{{\partial ^k }}{{\partial x_1^k ...\partial x_d^{k_d } }}\delta _\Gamma (B_t ), k_1 + ... + k_d = k > 1,\) of a d-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δΓ(·) is a generalization of the δ-function constructed on a bounded closed smooth surface Γ⊂R d , k≥1 and acting on finite continuous functions φ(·) in R d according to the rule \((\delta _\Gamma ,\varphi ) : = \int\limits_\Gamma {\varphi (x} )\lambda (dx),\) where ι(·) is a surface measure on Γ. |
| first_indexed | 2026-03-24T02:58:31Z |
| format | Article |
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| id | umjimathkievua-article-4405 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:58:31Z |
| publishDate | 2000 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/79/532080922c21a44af1a849d528b48779.pdf |
| spelling | umjimathkievua-article-44052020-03-18T20:28:32Z On generalized local time for the process of brownian motion Об обобщенном локальном времени для процесса броуновского движения Вакип, V. V. Бакун, В. В. Бакун, В. В. We prove that the functionals \(\delta _\Gamma (B_t ) and \frac{{\partial ^k }}{{\partial x_1^k ...\partial x_d^{k_d } }}\delta _\Gamma (B_t ), k_1 + ... + k_d = k > 1,\) of a d-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δΓ(·) is a generalization of the δ-function constructed on a bounded closed smooth surface Γ⊂R d , k≥1 and acting on finite continuous functions φ(·) in R d according to the rule \((\delta _\Gamma ,\varphi ) : = \int\limits_\Gamma {\varphi (x} )\lambda (dx),\) where ι(·) is a surface measure on Γ. Доводиться, що функціонали $\delta _\Gamma (B_t )$ та $\frac{{\partial ^k }}{{\partial x_1^k ...\partial x_d^{k_d } }}\delta _\Gamma (B_t ), k_1 + ... + k_d = k > 1,$ від $d$-мірного процесу броунівського руху є Хіда-розподілами, тобто узагальненими вінеровими функціоалами, де $δ_{Γ}(·)$ —узагальнення $δ$-функції, яке побудовано по обмеженій замкненій гладкій поверхні $Γ ⊂ R^d,\; k ≥ 1$, і визначене дією па фінітні неперервні функції $φ(·)$ в $R^d$ за правилом $(\delta _\Gamma ,\varphi ) : = \int\limits_\Gamma {\varphi (x} )\lambda (dx),$ де $ι(·)$ — поверхнева міра на $Γ$. Institute of Mathematics, NAS of Ukraine 2000-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4405 Ukrains’kyi Matematychnyi Zhurnal; Vol. 52 No. 2 (2000); 157-164 Український математичний журнал; Том 52 № 2 (2000); 157-164 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4405/5514 https://umj.imath.kiev.ua/index.php/umj/article/view/4405/5515 Copyright (c) 2000 Вакип V. V. |
| spellingShingle | Вакип, V. V. Бакун, В. В. Бакун, В. В. On generalized local time for the process of brownian motion |
| title | On generalized local time for the process of brownian motion |
| title_alt | Об обобщенном локальном времени для процесса броуновского движения |
| title_full | On generalized local time for the process of brownian motion |
| title_fullStr | On generalized local time for the process of brownian motion |
| title_full_unstemmed | On generalized local time for the process of brownian motion |
| title_short | On generalized local time for the process of brownian motion |
| title_sort | on generalized local time for the process of brownian motion |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4405 |
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