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Uspekhi Mat. Nauk, 2003, Volume 58, Issue 3(351), Pages 3–50 (Mi umn627)  

This article is cited in 23 scientific papers (total in 24 papers)

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

A. M. Vershik, N. V. Tsilevich

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/rm627

Full text: PDF file (603 kB)
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English version:
Russian Mathematical Surveys, 2003, 58:3, 427–472

Bibliographic databases:

UDC: 519.21+517.98
MSC: Primary 60G51, 46B28; Secondary 60G55, 60H40, 47A67, 47B34, 81R10, 28C20, 33C45
Received: 27.02.2003

Citation: A. M. Vershik, N. V. Tsilevich, “Fock factorizations, and decompositions of the $L^2$ spaces over general Lévy processes”, Uspekhi Mat. Nauk, 58:3(351) (2003), 3–50; Russian Math. Surveys, 58:3 (2003), 427–472

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Anshelevich M., “$q$-Lévy processes”, J. Reine Angew. Math., 576 (2004), 181–207  crossref  mathscinet  zmath  isi
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    3. Huang Zhiyuan, Wu Ying, “Lévy white noise calculus based on interaction exponents”, Acta Appl. Math., 88:3 (2005), 251–268  crossref  mathscinet  zmath  isi  elib  scopus
    4. Graev M.I., Vershik A.M., “The basic representation of the current group $\mathrm{O}(n,1)^X$ in the $L^2$ space over the generalized Lebesgue measure”, Indag. Math. (N.S.), 16:3-4 (2005), 499–529  crossref  mathscinet  zmath  isi  elib  scopus
    5. Berezansky Yu.M., Pulemyotov A.D., “Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field”, Ukrainian Math. J., 59:6 (2007), 811–832  crossref  mathscinet  zmath  elib  scopus
    6. A. M. Vershik, “Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space?”, Proc. Steklov Inst. Math., 259 (2007), 248–272  mathnet  crossref  mathscinet  zmath  elib  elib
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    15. Petlenko A.V., Kopytenko Yu.A., “Influence of Anomalous Particle Diffusion on the Current System Formation and Coherence of Pi2 Geomagnetic Pulsation Local Ionospheric Sources”, Geomagn. Aeron., 55:1 (2015), 24–31  crossref  isi  scopus
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    18. Mirzaee F., Hadadian E., “A New Computational Method For Solving Two-Dimensional Stratonovich Volterra Integral Equation”, Math. Meth. Appl. Sci., 40:16 (2017), 5777–5791  crossref  mathscinet  zmath  isi  scopus
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