In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the H-derivative is a notion of derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 in the study of abstract Wiener space

Abstract Wiener space

An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...

s and the Malliavin calculus

Malliavin calculus

The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables....

.

Definition

Let be an abstract Wiener space, and suppose that is differentiable

Fréchet derivative

In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...

. Then the Fréchet derivative

Fréchet derivative

In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...

 is a map;
i.e., for , is an element of , the dual space

Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

 to .

Therefore, define the -derivative at by,
a continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 linear map on .

Define the -gradient by.
That is, if denotes the adjoint

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 = .Specifically, adjoint may mean:...

of , we have .