H-derivative
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In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.[1]
Definition
[edit]Let be an abstract Wiener space, and suppose that is differentiable. Then the Fréchet derivative is a map
- ;
i.e., for , is an element of , the dual space to .
Therefore, define the -derivative at by
- ,
a continuous linear map on .
Define the -gradient by
- .
That is, if denotes the adjoint of , we have .
See also
[edit]References
[edit]- ^ 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.