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 Trudy Mat. Inst. Steklova, 2018, Volume 301, Pages 18–32 (Mi tm3901)

Lévy Laplacians in Hida calculus and Malliavin calculus

B. O. Volkovab

a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
b Bauman Moscow State Technical University, Vtoraya Baumanskaya ul. 5/1, Moscow, 105005 Russia

Abstract: Some connections between different definitions of Lévy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev–Schwartz distributions over the Wiener measure (Hida calculus). One can consider the chain of Lévy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Lévy Laplacian. Another approach to defining the Lévy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (Malliavin calculus). It is proved that the Lévy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Lévy Laplacians, but not with the classical Lévy Laplacian, under the embedding of the Sobolev space over the Wiener measure in the space of generalized functionals over this measure. It is shown which Lévy Laplacian in the stochastic analysis is connected with the gauge fields.

DOI: https://doi.org/10.1134/S0371968518020024

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 301, 11–24

Bibliographic databases:

UDC: 517.98

Citation: B. O. Volkov, “Lévy Laplacians in Hida calculus and Malliavin calculus”, Complex analysis, mathematical physics, and applications, Collected papers, Trudy Mat. Inst. Steklova, 301, MAIK Nauka/Interperiodica, Moscow, 2018, 18–32; Proc. Steklov Inst. Math., 301 (2018), 11–24

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3901
• https://doi.org/10.1134/S0371968518020024
• http://mi.mathnet.ru/eng/tm/v301/p18

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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. B. O. Volkov, “Lévy Laplacians and annihilation process”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 160, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2018, 399–409
2. 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
3. B. O. Volkov, “Levy Laplacian on Manifold and Yang-Mills Heat Flow”, Lobachevskii J. Math., 40:10, SI (2019), 1619–1630
4. B. O. Volkov, “Levy Laplacians and instantons on manifolds”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 23:2 (2020), 2050008
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