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Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, Volume 160, Book 2, Pages 399–409 (Mi uzku1466)  

This article is cited in 1 scientific paper (total in 1 paper)

Lévy Laplacians and annihilation process

B. O. Volkovab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991 Russia
b Bauman Moscow State Technical University, Moscow, 105005 Russia

Abstract: The Lévy Laplacians are infinite-dimensional Laplace operators defined as the Cesaro mean of the second-order directional derivatives. In the theory of Sobolev–Schwarz distributions over a Gaussian measure on an infinite-dimensional space (the Hida calculus), we can consider two canonical Lévy Laplacians. The first Laplacian, the so-called classical Lévy Laplacian, has been well studied. The interest in the second Laplacian is due to its connection with the Malliavin calculus (the theory of Sobolev spaces over the Wiener measure) and the Yang–Mills gauge theory. The representation in the form of the quadratic function of the annihilation process for the classical Lévy-Laplacian is known. This representation can be obtained using the $S$-transform (the Segal–Bargmann transform). In the paper, we show, by analogy, that the representation in the form of the quadratic function of the derivative of the annihilation process exists for the second Lévy-Laplacian. The obtained representation can be used for studying the gauge fields and the Lévy Laplacian in the Malliavin calculus.

Keywords: Lévy Laplacian, Hida calculus, quantum probability, annihilation process.

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Bibliographic databases:
UDC: 517.9
Received: 19.12.2017
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Citation: B. O. Volkov, “Lévy Laplacians and annihilation process”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 160, no. 2, Kazan University, Kazan, 2018, 399–409

Citation in format AMSBIB
\Bibitem{Vol18}
\by B.~O.~Volkov
\paper L\'evy Laplacians and annihilation process
\serial Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
\yr 2018
\vol 160
\issue 2
\pages 399--409
\publ Kazan University
\publaddr Kazan
\mathnet{http://mi.mathnet.ru/uzku1466}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3915727}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000460032400022}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. B. O. Volkov, “Levy differential operators and gauge invariant equations for Dirac and Higgs fields”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 22:1 (2019), 1950001  crossref  mathscinet  zmath  isi  scopus
  • Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
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