RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Uspekhi Mat. Nauk: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

 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)
References: PDF file   HTML file

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
\Bibitem{VerTsi03} \by A.~M.~Vershik, N.~V.~Tsilevich \paper Fock factorizations, and decompositions of the $L^2$ spaces over general L\'evy processes \jour Uspekhi Mat. Nauk \yr 2003 \vol 58 \issue 3(351) \pages 3--50 \mathnet{http://mi.mathnet.ru/umn627} \crossref{https://doi.org/10.4213/rm627} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1998773} \zmath{https://zbmath.org/?q=an:1060.46056} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2003RuMaS..58..427V} \elib{http://elibrary.ru/item.asp?id=13421645} \transl \jour Russian Math. Surveys \yr 2003 \vol 58 \issue 3 \pages 427--472 \crossref{https://doi.org/10.1070/RM2003v058n03ABEH000627} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000186019600001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0242298664} 

Linking options:
• http://mi.mathnet.ru/eng/umn627
• https://doi.org/10.4213/rm627
• http://mi.mathnet.ru/eng/umn/v58/i3/p3

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Anshelevich M., “$q$-Lévy processes”, J. Reine Angew. Math., 576 (2004), 181–207
2. Neretin Yu.A., “Structures of boson and fermion Fock spaces in the space of symmetric functions”, Acta Appl. Math., 81:1 (2004), 233–268
3. Huang Zhiyuan, Wu Ying, “Lévy white noise calculus based on interaction exponents”, Acta Appl. Math., 88:3 (2005), 251–268
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
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
6. A. M. Vershik, “Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space?”, Proc. Steklov Inst. Math., 259 (2007), 248–272
7. A. M. Vershik, M. I. Graev, “Integral Models of Representations of Current Groups”, Funct. Anal. Appl., 42:1 (2008), 19–27
8. A. M. Vershik, M. I. Graev, “Integral Models of Unitary Representations of Current Groups with Values in Semidirect Products”, Funct. Anal. Appl., 42:4 (2008), 279–289
9. Vershik A.M., “Invariant measures for the continual Cartan subgroup”, J. Funct. Anal., 255:9 (2008), 2661–2682
10. A. M. Vershik, M. I. Graev, “Integral models of representations of the current groups of simple Lie groups”, Russian Math. Surveys, 64:2 (2009), 205–271
11. Farre M., Jolis M., Utzet F., “Multiple Stratonovich Integral and Hu-Meyer Formula for Levy Processes”, Annals of Probability, 38:6 (2010), 2136–2169
12. A. M. Vershik, M. I. Graev, “Poisson model of the Fock space and representations of current groups”, St. Petersburg Math. J., 23:3 (2012), 459–510
13. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179
14. Accardi L., Rebei H., Riahi A., “The Quantum Decomposition Associated With the Levy White Noise Processes Without Moments”, Prob. Math. Stat.., 34:2 (2014), 337–362
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
16. M. Bożejko, E. W. Lytvynov, I. V. Rodionova, “An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions”, Russian Math. Surveys, 70:5 (2015), 857–899
17. Frei M.M. Kachanovsky N.A., “Some Remarks on Operators of Stochastic Differentiation in the Levy White Noise Analysis”, Methods Funct. Anal. Topol., 23:4 (2017), 320–345
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
19. Kachanovsky N.A., “On Wick Calculus on Spaces of Nonregular Generalized Functions of Levy White Noise Analysis”, Carpathian Math. Publ., 10:1 (2018), 114–132
20. Frei M.M., “Wick Calculus on Spaces of Regular Generalized Functions of Levy White Noise Analysis”, Carpathian Math. Publ., 10:1 (2018), 82–104
21. Kachanovsky N.A. Kachanovska T.O., “Interconnection Between Wick Multiplication and Integration on Spaces of Nonregular Generalized Functions in the Levy White Noise Analysis”, Carpathian Math. Publ., 11:1 (2019), 70–88
22. Dello Schiavo L., “Characteristic Functionals of Dirichlet Measures”, Electron. J. Probab., 24 (2019), 115
23. Frei M.M., Kachanovsky N.A., “on the Relationship Between Wick Calculus and Stochastic Integration in the Levy White Noise Analysis”, Eur. J. Math., 6:1, SI (2020), 179–196
24. Keys D. Wehr J., “Poisson Stochastic Master Equation Unravelings and the Measurement Problem: a Quantum Stochastic Calculus Perspective”, J. Math. Phys., 61:3 (2020), 032101
•  Number of views: This page: 681 Full text: 337 References: 59 First page: 6

 Contact us: math-net2020_10 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020