H-derivative

In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.^[1]

Definition

Let i : H \to E be an abstract Wiener space, and suppose that F : E \to \mathbb{R} is differentiable. Then the Fréchet derivative is a map

\mathrm{D} F : E \to \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 E by

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

Define the H-gradient \nabla_{H} F : E \to H by

\langle \nabla_{H} F (x), h \rangle_{H} = \left( \mathrm{D}_{H} F \right) (x) (h) = \lim_{t \to 0} \frac{F (x + t i(h)) - F(x)}{t}

That is, if j : E^{*} \to H denotes the adjoint of i : H \to E, we have \nabla_{H} F (x) := j \left( \mathrm{D} F (x) \right).

^ Victor Kac & Pokman Cheung (2002). Quantum Calculus. New York: Springer. pp. 80–84. doi:10.1007/978-1-4613-0071-7. ISBN 978-1-4613-0071-7.