Universal Malliavin Calculus in Fock and L\'{e}vy-It\^{o} 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 $L^2$-equivalence of norms is proved and an abstract version of the It^{o}-Skorohod isometry is established. We then outline a new proof of It^{o}'s chaos expansion of complex L'{e}vy-It^{o} space in terms of multiple Wiener-L'{e}vy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform now identifies L'{e}vy-It^{o} space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general L'{e}vy process. In particular we establish maximal domains of these operators and obtain the It^{o}-Skorohod isometry on its maximal domain.