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Fock factorizations and decompositions of the \(L^ 2\) spaces over general Lévy processes. (English. Russian original) Zbl 1060.46056

Russ. Math. Surv. 58, No. 3, 427-472 (2003); translation from Usp. Mat. Nauk 58, No. 3, 3-50 (2003).
Summary: 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.

MSC:

46N30 Applications of functional analysis in probability theory and statistics
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46G99 Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G51 Processes with independent increments; Lévy processes
60H40 White noise theory
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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