H-derivative

In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.¹

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

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

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

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

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

a continuous linear map on ${\displaystyle H}$.

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

${\displaystyle \langle \nabla _{H}F(x),h\rangle _{H}=\left(\mathrm {D} _{H}F\right)(x)(h)=\lim _{t\to 0}{\frac {F(x+ti(h))-F(x)}{t}}}$.

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