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$$\mu_t(B)=\int\limits _{[0,t]^p}1_B\big(X_1(s_1) - X_2(s_2),\ldots, X_{p-1}(s_{p-1}) -$$ $$- X_p(s_p)\b...
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| Дата: | 2020 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | Українська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2020
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/6278 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | UDC 519.21
Given the i.i.d. $\mathbb{R}^d$-valued stochastic processes $X_1(t),\ldots, X_p(t),$ $p\ge 2,$ with the stationary increments, a minimal condition is provided for the occupation measure$$\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. |
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| DOI: | 10.37863/umzh.v72i9.6278 |