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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Random dynamical system ： ウィキペディア英語版 Random dynamical system In the mathematical field of dynamical systems, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are characterized by a state space ''S'', a set of maps ''T'' from ''S'' into itself that can be thought of as the set of all possible equations of motion, and a probability distribution ''Q'' on the set ''T'' that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state $X\; \backslash in\; S$ evolving according to a succession of maps randomly chosen according to the distribution ''Q''. An example of a random dynamical system is a stochastic differential equation; in this case the distribution Q is typically determined by ''noise terms''. It consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. ==Motivation: solutions to a stochastic differential equation== Let $f\; :\; \backslash mathbb^\; \backslash to\; \backslash mathbb^$ be a $d$-dimensional vector field, and let $\backslash varepsilon\; >\; 0$. Suppose that the solution $X(t,\; \backslash omega;\; x\_)$ to the stochastic differential equation :$\backslash left\backslash \; X\; =\; f(X)\; \backslash ,\; \backslash mathrm\; t\; +\; \backslash varepsilon\; \backslash ,\; \backslash mathrm\; W\; (t);\; \backslash \backslash \; X\; (0)\; =\; x\_;\; \backslash end\; \backslash right.$ exists for all positive time and some (small) interval of negative time dependent upon $\backslash omega\; \backslash in\; \backslash Omega$, where $W\; :\; \backslash mathbb\; \backslash times\; \backslash Omega\; \backslash to\; \backslash mathbb^$ denotes a $d$-dimensional Wiener process (Brownian motion). Implicitly, this statement uses the classical Wiener probability space :$(\backslash Omega,\; \backslash mathcal,\; \backslash mathbb)\; :=\; \backslash left(\; C\_\; (\backslash mathbb;\; \backslash mathbb^),\; \backslash mathcal\; (C\_\; (\backslash mathbb;\; \backslash mathbb^)),\; \backslash gamma\; \backslash right).$ In this context, the Wiener process is the coordinate process. Now define a flow map or (solution operator) $\backslash varphi\; :\; \backslash mathbb\; \backslash times\; \backslash Omega\; \backslash times\; \backslash mathbb^\; \backslash to\; \backslash mathbb^$ by :$\backslash varphi\; (t,\; \backslash omega,\; x\_)\; :=\; X(t,\; \backslash omega;\; x\_)$ (whenever the right hand side is well-defined). Then $\backslash varphi$ (or, more precisely, the pair $(\backslash mathbb^,\; \backslash varphi)$) is a (local, left-sided) random dynamical system. The process of generating a "flow" from the solution to a stochastic differential equation leads us to study suitably defined "flows" on their own. These "flows" are random dynamical systems. 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「Random dynamical system」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.