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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]


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]


  1. ^ Victor Kac; Pokman Cheung (2002). Quantum Calculus. New York: Springer. pp. 80–84. ISBN 978-1-4613-0071-7.