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

$\mathrm{D}F:E\to \mathrm{L}\mathrm{i}\mathrm{n}(E;\mathbb{ℝ})$;

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

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

a continuous linear map on $H$.

Define the $H$-gradient ${\mathrm{\nabla }}_{H}F:E\to H$ by

$⟨{\mathrm{\nabla }}_{H}F(x),h{⟩}_H=\left({\mathrm{D}}_{H}F\right)(x)(h)=\underset{t\to 0}{\lim }\frac{F(x+ti(h))-F(x)}{t}$.

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

See also

• Malliavin derivative