Английская Википедия:H-derivative

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Шаблон:Technical Шаблон:Ref improve Шаблон:Use dmy dates In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.

Definition

Let <math>i : H \to E</math> be an abstract Wiener space, and suppose that <math>F : E \to \mathbb{R}</math> is differentiable. Then the Fréchet derivative is a map

<math>\mathrm{D} F : E \to \mathrm{Lin} (E; \mathbb{R})</math>;

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>.

Therefore, define the <math>H</math>-derivative <math>\mathrm{D}_{H} F</math> at <math>x \in E</math> by

<math>\mathrm{D}_{H} F (x) := \mathrm{D} F (x) \circ i : H \to \R</math>,

a continuous linear map on <math>H</math>.

Define the <math>H</math>-gradient <math>\nabla_{H} F : E \to H</math> by

<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>.

That is, if <math>j : E^{*} \to H</math> denotes the adjoint of <math>i : H \to E</math>, we have <math>\nabla_{H} F (x) := j \left( \mathrm{D} F (x) \right)</math>.

See also

• Malliavin derivative

References

Шаблон:Probability-stub