Universal Malliavin Calculus in Fock and Lévy-Itô Spaces

We review and extend Lindsay’s work on abstract gradient and divergence operators in Fock space over a general complex Hilbert space. Precise expressions for the domains are given, the L2-equivalence of norms is proved and an abstract version of the Itô-Skorohod isometry is established. We then outline a new proof of Itô’s chaos expansion of complex Lévy-Itô space in terms of multiple Wiener-Lévy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform now identifies Lévy-Itô space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general Lévy process. In particular we establish maximal domains of these operators and obtain the Itô-Skorohod isometry on its maximal domain.