翻訳と辞書 Words near each other ・ Borel (crater) ・ Borel (surname) ・ Borel Bo.11 ・ Borel conjecture ・ Borel determinacy theorem ・ Borel distribution ・ Borel equivalence relation ・ Borel fixed-point theorem ・ Borel functional calculus ・ Borel hierarchy ・ Borel hydro-monoplane ・ Borel isomorphism ・ Borel measure ・ Borel military monoplane ・ Borel regular measure ・ Borel right process ・ Borel set ・ Borel subgroup ・ Borel summation ・ Borel Torpille ・ Borel transform ・ Borel's lemma ・ Borel's theorem ・ Borel, California ・ Borel-Odier Bo-T ・ Boreland ・ Borella ・ Borella (game) ・ Borella Electoral District ・ Borelli Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Borel right process ： ウィキペディア英語版 Borel right process In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process. Let $E$ be a locally compact, separable, metric space. We denote by $\backslash mathcal\; E$ the Borel subsets of $E$. Let $\backslash Omega$ be the space of right continuous maps from $,\; B\; \backslash in\; \backslash mathcal\; E^$ *\right\}, and then, let : $$ \mathcal F_\infty = \sigma\left\(B) : s\in and the mapping given by $t\; \backslash rightarrow\; X\_t$ is right continuous, we see that for any uniformly continuous function $f$, we have the mapping given by $t\; \backslash rightarrow\; P\_tf(x)$ is right continuous. Therefore, together with the monotone class theorem, for any universally measurable function $f$, the mapping given by $(t,x)\; \backslash rightarrow\; P\_tf(x)$, is jointly measurable, that is, $\backslash mathcal\; B([0,\backslash infty))\backslash otimes\; \backslash mathcal\; E^$ * measurable, and subsequently, the mapping is also $\backslash left(\backslash mathcal\; B([0,\backslash infty))\backslash otimes\; \backslash mathcal\; E^$ *\right)^-measurable for all finite measures $\backslash lambda$ on $\backslash mathcal\; B([0,\backslash infty))$ and $\backslash mu$ on $\backslash mathcal\; E^$ *. Here, $\backslash left(\backslash mathcal\; B([0,\backslash infty))\backslash otimes\; \backslash mathcal\; E^$ *\right)^ is the completion of $\backslash mathcal\; B([0,\backslash infty))\backslash otimes\; \backslash mathcal\; E^$ * with respect to the product measure $\backslash lambda\; \backslash otimes\; \backslash mu$. Thus, for any bounded universally measurable function $f$ on $E$, the mapping $t\backslash rightarrow\; P\_tf(x)$ is Lebeague measurable, and hence, for each $\backslash alpha\; \backslash in\; [0,\backslash infty)$, one can define : $$ U^\alpha f(x) = \int_0^\infty e^P_tf(x) dt. There is enough joint measurability to check that $\backslash$ is a Markov resolvent on $(E,\backslash mathcal\; E^$ *), which uniquely associated with the Markovian semigroup $\backslash$. Consequently, one may apply Fubini's theorem to see that : $$ U^\alpha f(x) = \mathbf E^x\left[ \int_0^\infty e^ f(X_t) dt \right]. The followings are the defining properties of Borel right processes: * Hypothesis Droite 1: :For each probability measure $\backslash mu$ on $(E,\; \backslash mathcal\; E)$, there exists a probability measure $\backslash mathbf\; P^\backslash mu$ on $(\backslash Omega,\; \backslash mathcal\; F^$ *) such that $(X\_t,\; \backslash mathcal\; F\_t^$ *, P^\mu) is a Markov process with initial measure $\backslash mu$ and transition semigroup $\backslash$. * Hypothesis Droite 2: :Let $f$ be $\backslash alpha$-excessive for the resolvent on $(E,\; \backslash mathcal\; E^$ *). Then, for each probability measure $\backslash mu$ on $(E,\backslash mathcal\; E)$, a mapping given by $t\; \backslash rightarrow\; f(X\_t)$ is $P^\backslash mu$ almost surely right continuous on $[0,\backslash infty)$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Borel right process」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.