H-derivative

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

[edit] Definition

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

$\mathrm{D} F : E \to \mathrm{Lin} (E; \mathbb{R})$ ;

i.e., for $x \in E$ , $D F (x)$ is an element of $E *$ , the dual space to $E$ .

Therefore, define the $H$ -derivative $D H F$ at $x \in X$ by

$\mathrm{D}_{H} F (x) := \mathrm{D} F (x) \circ i : H \to \R$ ,

Define the $H$ -gradient $\nabla_{H} F : E \to H$ by

$\langle \nabla_{H} F (x), h \rangle_{H} = \left( \mathrm{D}_{H} F \right) (x) (h) = \lim_{t \to 0} \frac{F (x + t i(h)) - F(x)}{t}$ .

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