Fock factorizations, and decompositions of the $L^2$ spaces over general L\'evy processes

A. M. Vershik ; N. V. Tsilevich

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 3, pp. 427-472

Résumé

This paper is devoted to an explicit construction and study of an isometry between the spaces of square-integrable functionals of an arbitrary Lévy process (a process with independent values) and of a vector-valued Gaussian white noise. Explicit formulae are obtained for this isometry on the level of multiplicative functionals and orthogonal decompositions. The central special case is treated at length, that is, the case of an isometry between the $L^2$ spaces over a Poisson process and over a white noise; in particular, an explicit combinatorial formula is given for the kernel of this isometry. A key role in our considerations is played by the concepts of measure factorization and Hilbert factorization, as well as the closely related concepts of multiplicative and additive functionals and of taking the logarithm in factorizations. The results obtained make possible the introduction of a canonical Fock structure (an analogue of the Wiener–Itô decomposition) in the $L^2$ space over an arbitrary Lévy process. Applications to the theory of representations of current groups are also considered, and an example of a non-Fock factorization is given.