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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 Freidlin–Wentzell theorem ： ウィキペディア英語版 Freidlin–Wentzell theorem In mathematics, the Freidlin–Wentzell theorem is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentzell theorem gives an estimate for the probability that a (scaled-down) sample path of an Itō diffusion will stray far from the mean path. This statement is made precise using rate functions. The Freidlin–Wentzell theorem generalizes Schilder's theorem for standard Brownian motion. ==Statement of the theorem== Let ''B'' be a standard Brownian motion on R''d'' starting at the origin, 0 ∈ R''d'', and let ''X''''ε'' be an R''d''-valued Itō diffusion solving an Itō stochastic differential equation of the form :$\backslash begin\; \backslash mathrm\; X\_^\; =\; b(X\_^)\; \backslash ,\; \backslash mathrm\; t\; +\; \backslash sqrt\; \backslash ,\; \backslash mathrm\; B\_;\; \backslash \backslash \; X\_^\; =\; 0;\; \backslash end$ where the drift vector field ''b'' : R''d'' → R''d'' is uniformly Lipschitz continuous. Then, on the Banach space ''C''0 = ''C''0((); R''d'') equipped with the supremum norm ||·||∞, the family of processes (''X''''ε'')''ε''>0 satisfies the large deviations principle with good rate function ''I'' : ''C''0 → R ∪  given by :$I(\backslash omega)\; =\; \backslash frac\; \backslash int\_0^T\; |\; \backslash dot\_t\; -\; b(\backslash omega\_t)\; |^2\; \backslash ,\; \backslash mathrm\; t$ if ''ω'' lies in the Sobolev space ''H''1((); R''d''), and ''I''(''ω'') = +∞ otherwise. In other words, for every open set ''G'' ⊆ ''C''0 and every closed set ''F'' ⊆ ''C''0, :$\backslash limsup\_\; \backslash varepsilon\; \backslash log\; \backslash mathbf\; \backslash big(X^\; \backslash in\; F\; \backslash big)\; \backslash leq\; -\; \backslash inf\_\; I(\backslash omega)$ and :$\backslash liminf\_\; \backslash varepsilon\; \backslash log\; \backslash mathbf\; \backslash big(X^\; \backslash in\; G\; \backslash big)\; \backslash geq\; -\; \backslash inf\_\; I(\backslash omega).$ 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「Freidlin–Wentzell theorem」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.