Summary: Universal Malliavin Calculus in Fock and
Probability and Statistics Department,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
We review and extend Lindsay's work on abstract gradient and diver-
gence operators in Fock space over a general complex Hilbert space. Pre-
cise expressions for the domains are given, the L2
-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´evy-It^o
space in terms of multiple Wiener-L´evy integrals based on Brownian mo-
tion and a compensated Poisson random measure. The duality transform
now identifies L´evy-It^o space as a Fock space. We can then easily obtain
key properties of the gradient and divergence of a general L´evy process.
In particular we establish maximal domains of these operators and obtain