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 $i:H\to E$ 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 $x \in E$ , $\mathrm {D} F(x)$ is an element of $E^{{*}}$ , the dual space to $E$ .

Therefore, define the $H$ -derivative $\mathrm {D} _{H}F$ at $x \in E$ by

\mathrm {D} _{H}F(x):=\mathrm {D} F(x)\circ i:H\to \mathbb {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+ti(h))-F(x)}{t}}

.

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