Fock factorizations, and decompositions of the $L^2$ spaces over general Levy processes

Details Fock factorizations, and decompositions of the $L^2$ spaces over general Levy processes Fock factorizations, and decompositions of the $L^2$ spaces over general Levy processes

2003-04-20

arxiv.org/abs/math/0304282v1

Mathematics

Probability

35 pages; LaTeX; to appear in Russian Math. Surveys

Scientific paper

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 $L^2$ 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 $L^2$ 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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