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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 Schilder's theorem ： ウィキペディア英語版 Schilder's theorem In mathematics, Schilder's theorem is a result in the large deviations theory of stochastic processes. Roughly speaking, Schilder's theorem gives an estimate for the probability that a (scaled-down) sample path of Brownian motion will stray far from the mean path (which is constant with value 0). This statement is made precise using rate functions. Schilder's theorem is generalized by the Freidlin–Wentzell theorem for Itō diffusions. ==Statement of the theorem== Let ''B'' be a standard Brownian motion in ''d''-dimensional Euclidean space R''d'' starting at the origin, 0 ∈ R''d''; let W denote the law of ''B'', i.e. classical Wiener measure. For ''ε'' > 0, let W''ε'' denote the law of the rescaled process (√''ε'')''B''. Then, on the Banach space ''C''0 = ''C''0((); R''d'') of continuous functions $f\; :\; ()\; \backslash longrightarrow\; \backslash mathbf^d$ such that $f(0)=0$, equipped with the supremum norm ||·||∞, the probability measures W''ε'' satisfy the large deviations principle with good rate function ''I'' : ''C''0 → R ∪  given by :$I(\backslash omega)\; =\; \backslash frac\; \backslash int\_^\; |\; \backslash dot(t)\; |^\; \backslash ,\; \backslash mathrm\; t$ if ''ω'' is absolutely continuous, and ''I''(''ω'') = +∞ otherwise. In other words, for every open set ''G'' ⊆ ''C''0 and every closed set ''F'' ⊆ ''C''0, :$\backslash limsup\_\; \backslash varepsilon\; \backslash log\; \backslash mathbf\_\; (F)\; \backslash leq\; -\; \backslash inf\_\; I(\backslash omega)$ and :$\backslash liminf\_\; \backslash varepsilon\; \backslash log\; \backslash mathbf\_\; (G)\; \backslash geq\; -\; \backslash inf\_\; I(\backslash omega).$ 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「Schilder's 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.