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Fock factorizations, and decompositions of the spaces over general Levy processes
We explicitly construct and study an 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 at the level of
orthogonal decompositions, as well as find its kernel. We consider in detail
the central special case: the isometry between the spaces over a Poisson
process and the corresponding white noise. 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 space over an arbitrary Levy process.
An application to the representation theory of current groups is considered. An
example of a non-Fock factorization is given.Comment: 35 pages; LaTeX; to appear in Russian Math. Survey
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