In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.
Let i : H → E be an abstract Wiener space, and suppose that F : E → R is differentiable. Then the Fréchet derivative is a map
D F : E → L i n ( E ; R ) ;
i.e., for x ∈ E , D F ( x ) is an element of E ∗ , the dual space to E .
Therefore, define the H -derivative D H F at x ∈ E by
D H F ( x ) := D F ( x ) ∘ i : H → R ,
a continuous linear map on H .
Define the H -gradient ∇ H F : E → H by
⟨ ∇ H F ( x ) , h ⟩ H = ( D H F ) ( x ) ( h ) = lim t → 0 F ( x + t i ( h ) ) − F ( x ) t .
That is, if j : E ∗ → H denotes the adjoint of i : H → E , we have ∇ H F ( x ) := j ( D F ( x ) ) .
