H-derivative

{{technical|date=June 2012}} {{Ref improve|date=February 2024}} {{Use dmy dates|date=February 2024}} In mathematics, the '''''H''-derivative''' is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.<ref>{{cite book|author1=Victor Kac|author2=Pokman Cheung |title=Quantum Calculus |year=2002 |publisher=Springer |location=New York |isbn=978-1-4613-0071-7 |pages=80–84 |doi=10.1007/978-1-4613-0071-7 |url=https://doi.org/10.1007/978-1-4613-0071-7}}</ref>

==Definition==

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

Therefore, define the '''<math>H</math>-derivative''' <math>\mathrm{D}_{H} F</math> at <math>x \in E</math> by :<math>\mathrm{D}_{H} F (x) := \mathrm{D} F (x) \circ i : H \to \R</math>, a continuous linear map on <math>H</math>.

Define the '''<math>H</math>-gradient''' <math>\nabla_{H} F : E \to H</math> by :<math>\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}</math>. That is, if <math>j : E^{*} \to H</math> denotes the adjoint of <math>i : H \to E</math>, we have <math>\nabla_{H} F (x) := j \left( \mathrm{D} F (x) \right)</math>.

==See also== * Malliavin derivative

==References== {{Reflist}}

Category:Generalizations of the derivative Category:Measure theory Category:Stochastic calculus