Uspekhi Mat. Nauk,

English translation: Russian Math. Surveys,

**Abstract.** 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. We obtain explicit
formulas for this isometry at the level of
multiplicative functionals and at the level of orthogonal
decompositions. We consider in detail the
central special case: the isometry between the L^2 spaces over
a Poisson process and the corresponding white noise; in particular,
we give an explicit combinatorial formula for the kernel of
this isometry.
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. An application to the representation
theory of current groups is considered. An example of a non-Fock
factorization is given.

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