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 → E {\displaystyle i:H\to E} be an abstract Wiener space, and suppose that F : E → R {\displaystyle F:E\to \mathbb {R} } is differentiable. Then the Fréchet derivative is a map
D F : E → L i n ( E ; R ) {\displaystyle \mathrm {D} F:E\to \mathrm {Lin} (E;\mathbb {R} )} ; i.e., for x ∈ E {\displaystyle x\in E} , D F ( x ) {\displaystyle \mathrm {D} F(x)} is an element of E ∗ {\displaystyle E^{*}} , the dual space to E {\displaystyle E} . Therefore, define the H {\displaystyle H} -derivative D H F {\displaystyle \mathrm {D} _{H}F} at x ∈ E {\displaystyle x\in E} by
D H F ( x ) := D F ( x ) ∘ i : H → R {\displaystyle \mathrm {D} _{H}F(x):=\mathrm {D} F(x)\circ i:H\to \mathbb {R} } , a continuous linear map on H {\displaystyle H} . Define the H {\displaystyle H} -gradient ∇ H F : E → H {\displaystyle \nabla _{H}F:E\to H} by
⟨ ∇ H F ( x ) , h ⟩ H = ( D H F ) ( x ) ( h ) = lim t → 0 F ( x + t i ( h ) ) − F ( x ) t {\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 j : E ∗ → H {\displaystyle j:E^{*}\to H} denotes the adjoint of i : H → E {\displaystyle i:H\to E} , we have ∇ H F ( x ) := j ( D F ( x ) ) {\displaystyle \nabla _{H}F(x):=j\left(\mathrm {D} F(x)\right)} .
See also Malliavin derivative
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