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| {{technical|date=June 2012}}
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| In [[mathematics]], the '''''H''-derivative''' is a notion of [[derivative]] in the study of [[abstract Wiener space]]s and the [[Malliavin calculus]].
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| ==Definition==
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| Let <math>i : H \to E</math> be an abstract Wiener space, and suppose that <math>F : E \to \mathbb{R}</math> is [[Fréchet_derivative|differentiable]]. Then the [[Fréchet derivative]] is a map
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| :<math>\mathrm{D} F : E \to \mathrm{Lin} (E; \mathbb{R})</math>;
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| i.e., for <math>x \in E</math>, <math>\mathrm{D} F (x)</math> is an element of <math>E^{*}</math>, the [[dual space]] to <math>E</math>.
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| Therefore, define the '''<math>H</math>-derivative''' <math>\mathrm{D}_{H} F</math> at <math>x \in E</math> by
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| :<math>\mathrm{D}_{H} F (x) := \mathrm{D} F (x) \circ i : H \to \R</math>,
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| a [[continuous function|continuous]] [[linear map]] on <math>H</math>.
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| Define the '''<math>H</math>-gradient''' <math>\nabla_{H} F : E \to H</math> by
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| :<math>\langle \nabla_{H} F (x), h \rangle_{H} = \left( \mathrm{D}_{H} F \right) (x) (h) = \lim_{t \to 0} \frac{F (x + t i(h)) - F(x)}{t}</math>.
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| That is, if <math>j : E^{*} \to H</math> denotes the [[adjoint]]{{dn|date=December 2013}} of <math>i : H \to E</math>, we have <math>\nabla_{H} F (x) := j \left( \mathrm{D} F (x) \right)</math>.
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| ==See also== | |
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| * [[Malliavin derivative]]
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| ==References==
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| {{unreferenced|date=June 2008}}
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| [[Category:Generalizations of the derivative]]
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| [[Category:Measure theory]]
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| [[Category:Stochastic calculus]]
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| {{probability-stub}}
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