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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 Onsager–Machlup function ： ウィキペディア英語版 Onsager–Machlup function The Onsager–Machlup function is a function that summarizes the dynamics of a continuous stochastic process. It is used to define a probability density for a stochastic process, and it is similar to the Lagrangian of a dynamical system. It is named after Lars Onsager and S. Machlup who were the first to consider such probability densities.〔Onsager, L. and Machlup, S. (1953)〕 The dynamics of a continuous stochastic process from time to in one dimension, satisfying a stochastic differential equation :$dX\_t\; =\; b(X\_t)\backslash ,dt\; +\; \backslash sigma(X\_t)\backslash ,dW\_t$ where is a Wiener process, can in approximation be described by the probability density function of its value at a finite number of points in time : :$p(x\_1,\backslash ldots,x\_n)\; =\; \backslash left(\; \backslash prod^\_\; \backslash frac\_\; L\backslash left(x\_i,\backslash frac\backslash right)\; \backslash ,\; \backslash Delta\; t\_i\; \backslash right)$ where :$L(x,v)\; =\; \backslash frac\backslash left(\backslash frac\backslash right)^2$ and , and . A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes , but in the limit the probability density function becomes ill defined, one reason being that the product of terms :$\backslash fract\backslash in()\; \backslash right)\}\; \backslash to\; \backslash exp\backslash left(-\backslash int^T\_0\; L\; \backslash left\; (\backslash varphi\_1(t),\backslash dot\_1(t)\; \backslash right\; )\; \backslash ,\; dt\; +\; \backslash int^T\_0\; L\; \backslash left\; (\backslash varphi\_2(t),\backslash dot\_2(t)\; \backslash right)\; \backslash ,\; dt\; \backslash right)$ as , where is the Onsager–Machlup function. ==Definition== Consider a -dimensional Riemannian manifold and a diffusion process on with infinitesimal generator , where is the Laplace–Beltrami operator and is a vector field. For any two smooth curves , :$\backslash lim\_\; \backslash frac\; =\; \backslash exp\backslash left(\; -\backslash int^T\_0\; L\; \backslash left\; (\backslash varphi\_1(t),\backslash dot\_1(t)\; \backslash right\; )\; \backslash ,\; dt\; +\backslash int^T\_0\; L\; \backslash left\; (\backslash varphi\_2(t),\backslash dot\_2(t)\; \backslash right\; )\; \backslash ,\; dt\; \backslash right)$ where is the Riemannian distance, $\backslash scriptstyle\; \backslash dot\_1,\; \backslash dot\_2$ denote the first derivatives of , and is called the Onsager–Machlup function. The Onsager–Machlup function is given by〔Takahashi, Y. and Watanabe, S. (1980)〕〔Fujita, T. and Kotani, S. (1982)〕〔Wittich, Olaf〕 :$L(x,v)\; =\; \backslash tfrac\backslash |v-b(x)\backslash |\_x^2\; +\backslash tfrac\backslash operatorname\backslash ,\; b(x)\; -\; \backslash tfracR(x),$ where is the Riemannian norm in the tangent space at , is the divergence of at , and is the scalar curvature at . 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「Onsager–Machlup function」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.