H-derivative

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 i : H o E be an abstract Wiener space, and suppose that F : E o mathbb{R} is differentiable. Then the Fréchet derivative is a
mathrm{D} F : E o mathrm{Lin} (E; mathbb{R});i.e., for x in E, mathrm{D} F (x) is an element of E^{*}, the dual space to E.

Therefore, define the H-derivative mathrm{D}_{H} F at x in X by:mathrm{D}_{H} F (x) := mathrm{D} F (x) circ i : H o R,a continuous linear map on H.

Define the H-gradient abla_{H} F : E o H by:langle abla_{H} F (x), h angle_{H} = left( mathrm{D}_{H} F ight) (x) (h) = lim_{t o 0} frac{F (x + t i(h)) - F(x)}{t}.That is, if j : E^{*} o H denotes the adjoint of i : H o E, we have abla_{H} F (x) := j left( mathrm{D} F (x) ight).

ee also

* Malliavin derivative

References


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