Steklov Institute of Mathematics at St.Petersburg

PREPRINT 23/2002

EXPANSIONS OF THE L^2 SPACES OVER GENERAL LEVY PROCESSES, AND FOCK FACTORIZATIONS

This preprint was accepted December 25, 2002
Contact: A. M. Vershik, N. Tsilevich

ABSTRACT:
We construct explicitly and study the isometry between the spaces
of square integrable functionals of an arbitrary Levy process and
a vector-valued Gaussian white noise. In particular, we obtain explicit 
formulas for this isometry at the level of 
multiplicative functionals and orthogonal
decompositions, and find its kernel.
The key role in our considerations is
played by the notion of measure and Hilbert factorizations
and related notions of multiplicative and additive functionals
and logarithm. The obtained
results allow us to introduce a canonical Fock structure
(an analogue of the Wiener-Ito decomposition) in the L^2 space
over an arbitrary Levy process.