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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 Hörmander's condition ： ウィキペディア英語版 Hörmander's condition In mathematics, Hörmander's condition is a property of vector fields that, if satisfied, has many useful consequences in the theory of partial and stochastic differential equations. The condition is named after the Swedish mathematician Lars Hörmander. ==Definition== Given two ''C''1 vector fields ''V'' and ''W'' on ''d''-dimensional Euclidean space R''d'', let denote their Lie bracket, another vector field defined by :$(W)\; (x)\; =\; \backslash mathrm\; V(x)\; W(x)\; -\; \backslash mathrm\; W(x)\; V(x),$ where D''V''(''x'') denotes the Fréchet derivative of ''V'' at ''x'' ∈ R''d'', which can be thought of as a matrix that is applied to the vector ''W''(''x''), and ''vice versa''. Let ''A''0, ''A''1, ... ''A''''n'' be vector fields on R''d''. They are said to satisfy Hörmander's condition if, for every point ''x'' ∈ R''d'', the vectors :$\backslash begin$ &A_ (x)~,\\ &((x), A_}" TITLE="A_} (x) ), A_ \qquad 0 \leq j_, j_, \ldots, j_ \leq n (span|span )] R''d''. They are said to satisfy the parabolic Hörmander condition if the same holds true, but with the index $j\_0$ taking only values in 1,...,n. Now consider the stochastic differential equation :$dx\; =\; V\_0(x)\; \backslash ;\; dt\; +\; \backslash sum\_^m\; V\_i(x)\; \backslash circ\; dW\_i$ where the vectors fields are assumed to have bounded derivative. Hörmander's theorem asserts that if the SDE above satisfies the parabolic Hörmander condition, then its solutions admit a smooth density with respect to Lebesgue measure. 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「Hörmander's condition」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.