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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 Reversible diffusion ： ウィキペディア英語版 Reversible diffusion In mathematics, a reversible diffusion is a specific example of a reversible stochastic process. Reversible diffusions have an elegant characterization due to the Russian mathematician Andrey Nikolaevich Kolmogorov. ==Kolmogorov's characterization of reversible diffusions== Let ''B'' denote a ''d''-dimensional standard Brownian motion; let ''b'' : R''d'' → R''d'' be a Lipschitz continuous vector field. Let ''X'' : [0, +∞) × Ω → R''d'' be an Itō diffusion defined on a probability space (Ω, Σ, P) and solving the Itō stochastic differential equation :$\backslash mathrm\; X\_\; =\; b(X\_)\; \backslash ,\; \backslash mathrm\; t\; +\; \backslash mathrm\; B\_$ with square-integrable initial condition, i.e. ''X''0 ∈ ''L''2(Ω, Σ, P; R''d''). Then the following are equivalent: * The process ''X'' is reversible with stationary distribution ''μ'' on R''d''. * There exists a scalar potential Φ : R''d'' → R such that ''b'' = −∇Φ, ''μ'' has Radon–Nikodym derivative ::$\backslash frac\; =\; \backslash exp\; \backslash left(\; -\; 2\; \backslash Phi\; (x)\; \backslash right)$ :and ::$\backslash int\_\}\; \backslash exp\; \backslash left(\; -\; 2\; \backslash Phi\; (x)\; \backslash right)\; \backslash ,\; \backslash mathrm\; x\; =\; 1.$ (Of course, the condition that ''b'' be the negative of the gradient of Φ only determines Φ up to an additive constant; this constant may be chosen so that exp(−2Φ(·)) is a probability density function with integral 1.) 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「Reversible diffusion」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.