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 \( {\displaystyle i:H\to E} \) 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 \( {\displaystyle \nabla _{H}F(x):=j\left(\mathrm {D} F(x)\right)} \).