H-derivative

In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.

Definition

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

[math]\displaystyle{ \mathrm{D} F : E \to \mathrm{Lin} (E; \mathbb{R}) }[/math];

i.e., for [math]\displaystyle{ x \in E }[/math], [math]\displaystyle{ \mathrm{D} F (x) }[/math] is an element of [math]\displaystyle{ E^{*} }[/math], the dual space to [math]\displaystyle{ E }[/math].

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

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

a continuous linear map on [math]\displaystyle{ H }[/math].

Define the [math]\displaystyle{ H }[/math]-gradient [math]\displaystyle{ \nabla_{H} F : E \to H }[/math] by

[math]\displaystyle{ \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]\displaystyle{ j : E^{*} \to H }[/math] denotes the adjoint of [math]\displaystyle{ i : H \to E }[/math], we have [math]\displaystyle{ \nabla_{H} F (x) := j \left( \mathrm{D} F (x) \right) }[/math].

See also

• Malliavin derivative