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 $i\; :\; H\; o\; E$ be an abstract Wiener space, and suppose that $F\; :\; E\; o\; mathbb\{R\}$ is differentiable. Then the Fréchet derivative is a
$mathrm\{D\}\; F\; :\; E\; o\; mathrm\{Lin\}\; (E;\; mathbb\{R\})$;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 $x\; in\; X$ by:$mathrm\{D\}\_\{H\}\; F\; (x)\; :=\; mathrm\{D\}\; F\; (x)\; circ\; i\; :\; H\; o\; R$,a continuous linear map on $H$.

Define the $H$-gradient $abla\_\{H\}\; F\; :\; E\; o\; H$ by:$langle\; abla\_\{H\}\; F\; (x),\; h\; angle\_\{H\}\; =\; left(\; mathrm\{D\}\_\{H\}\; F\; ight)\; (x)\; (h)\; =\; lim\_\{t\; o\; 0\}\; frac\{F\; (x\; +\; t\; i(h))\; -\; F(x)\}\{t\}$.That is, if $j\; :\; E^\{*\}\; o\; H$ denotes the adjoint of $i\; :\; H\; o\; E$, we have $abla\_\{H\}\; F\; (x)\; :=\; j\; left(\; mathrm\{D\}\; F\; (x)\; ight)$.

ee also

* Malliavin derivative

References