Definition
Let be the Cameron-Martin space, and denote classical Wiener space:
- ;
- ;
By the Sobolev embedding theorem, . Let
denote the inclusion map.
Suppose that is Fréchet differentiable. Then the Fréchet derivative is a map
- ;
i.e., for paths, is an element of, the dual space to . Denote by the continuous linear map defined by
sometimes known as the H-derivative. Now define to be the adjoint of in the sense that
- .
Then the Malliavin derivative is defined by
The domain of is the set of all Fréchet differentiable real-valued functions on ; the codomain is .
The Skorokhod integral is defined to be the adjoint of the Malliavin derivative:
Read more about this topic: Malliavin Derivative
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)