mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the ''H''-derivative is a notion of

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

in the study of

abstract Wiener space The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Camero ...

s and the

Malliavin calculus In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ...

Definition

Let

i : H \to E

be an abstract Wiener space, and suppose that

F : E \to \mathbb

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...

. Then the

Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued fu ...

is a map :

\mathrm F : E \to \mathrm (E; \mathbb)

; i.e., for

x \in E

\mathrm F (x)

is an element of

E^

, the

dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...

E

. Therefore, define the

H

-derivative

\mathrm_ F

x \in E

by :

\mathrm_ F (x) := \mathrm F (x) \circ i : H \to \R

, a

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...

H

. Define the

H

-gradient

\nabla_ F : E \to H

by :

\langle \nabla_ F (x), h \rangle_ = \left( \mathrm_ F \right) (x) (h) = \lim_ \frac

. That is, if

j : E^ \to H

denotes the

adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...

i : H \to E

, we have

\nabla_ F (x) := j \left( \mathrm F (x) \right)

References

Generalizations of the derivative Measure theory Stochastic calculus {{probability-stub

Definition

See also

References