Condition for intersection occupation measure to be absolutely continuous
UDC 519.21 Given the i.i.d. $\mathbb{R}^d$-valued stochastic processes $X_1(t),\ldots, X_p(t),$ $p\ge 2,$ with the stationary increments, a minimal condition is provided for the occupation measure$$\mu_t(B)=\int\limits _{[0,t]^p}1_B\big(X_1(s_1) - X_2(s_2),\ldots, X_{p-1}(s_{p-1}) -$$ $$- X_p(s_p)\b...
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| Datum: | 2020 |
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| Format: | Artikel |
| Sprache: | Ukrainisch |
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Institute of Mathematics, NAS of Ukraine
2020
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6278 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512308810219520 |
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| author | Chen, X. Chen, X. Chen, X. |
| author_facet | Chen, X. Chen, X. Chen, X. |
| author_sort | Chen, X. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:02:06Z |
| description | UDC 519.21
Given the i.i.d. $\mathbb{R}^d$-valued stochastic processes $X_1(t),\ldots, X_p(t),$ $p\ge 2,$ with the stationary increments, a minimal condition is provided for the occupation measure$$\mu_t(B)=\int\limits _{[0,t]^p}1_B\big(X_1(s_1) - X_2(s_2),\ldots, X_{p-1}(s_{p-1}) -$$
$$- X_p(s_p)\big)ds_1\ldots ds_p,\quad B\subset \mathbb{R}^{d(p-1)},$$to be absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^{d(p-1)}.$ An isometry identity related to the resulting density (known as intersection local time) is also established. |
| doi_str_mv | 10.37863/umzh.v72i9.6278 |
| first_indexed | 2026-03-24T03:26:44Z |
| format | Article |
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| id | umjimathkievua-article-6278 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:26:44Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-62782022-03-26T11:02:06Z Condition for intersection occupation measure to be absolutely continuous Условия абсолютной непрерывности меры посещения пересечений Condition for intersection occupation measure to be absolutely continuous Chen, X. Chen, X. Chen, X. Intersection local time occupation measure Plancherel-Parseval theorem Intersection local time occupation measure Plancherel-Parseval theorem UDC 519.21 Given the i.i.d. $\mathbb{R}^d$-valued stochastic processes $X_1(t),\ldots, X_p(t),$ $p\ge 2,$ with the stationary increments, a minimal condition is provided for the occupation measure$$\mu_t(B)=\int\limits _{[0,t]^p}1_B\big(X_1(s_1) - X_2(s_2),\ldots, X_{p-1}(s_{p-1}) -$$ $$- X_p(s_p)\big)ds_1\ldots ds_p,\quad B\subset \mathbb{R}^{d(p-1)},$$to be absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^{d(p-1)}.$ An isometry identity related to the resulting density (known as intersection local time) is also established. Для независимых одинаково распределенных R^d-значных случайных процессов X_1(t),..., X_p(t) (p>=2) со стационарными приращениями приведено минимальное условие, при котором мера посещения пересечений абсолютно непрерывна относительно меры Лебега на R^{d(p-1)}. Также доказано изометрическое тождество связанное с соответствующей плотностью (известной как локальное время пересечений).  УДК 519.21 Умова абсолютної неперервностi мiри вiдвiдувань перетинiв Для незалежних однаково розподілених $\mathbb{R}^d$-значних випадкових процесів $X_1(t),\ldots, X_p(t),$ $p\ge 2,$зі стаціонарними приростами наведено мінімальну умову, коли міра відвідувань перетинів $$\mu_t(B)=\int\limits _{[0,t]^p}1_B\big(X_1(s_1) - X_2(s_2),\ldots, X_{p-1}(s_{p-1}) -$$ $$- X_p(s_p)\big)ds_1\ldots ds_p,\quad B\subset \mathbb{R}^{d(p-1)},$$ абсолютно неперервна відносно міри Лебега на $\mathbb{R}^{d(p-1)}.$ Також доведено ізометричну тотожність, пов'язану із відповідною щільністю (відомою як локальний час перетинів). Institute of Mathematics, NAS of Ukraine 2020-09-22 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/6278 10.37863/umzh.v72i9.6278 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 9 (2020); 1304-1312 Український математичний журнал; Том 72 № 9 (2020); 1304-1312 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/6278/8756 Copyright (c) 2020 Xia Chen |
| spellingShingle | Chen, X. Chen, X. Chen, X. Condition for intersection occupation measure to be absolutely continuous |
| title | Condition for intersection occupation measure to be absolutely continuous |
| title_alt | Условия абсолютной непрерывности меры посещения пересечений Condition for intersection occupation measure to be absolutely continuous |
| title_full | Condition for intersection occupation measure to be absolutely continuous |
| title_fullStr | Condition for intersection occupation measure to be absolutely continuous |
| title_full_unstemmed | Condition for intersection occupation measure to be absolutely continuous |
| title_short | Condition for intersection occupation measure to be absolutely continuous |
| title_sort | condition for intersection occupation measure to be absolutely continuous |
| topic_facet | Intersection local time occupation measure Plancherel-Parseval theorem Intersection local time occupation measure Plancherel-Parseval theorem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6278 |
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