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

Definition

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

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

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

a continuous linear map on H.

Define the 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)}$$.

Malliavin derivative