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Quantum probability and Levy Laplacians

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Abstract

The formula Δ L = limɛ→0 ∫∥s–t∥<ɛ b s b t dsdt for the Levy Laplacian is obtained, where b t stands for an annihilation process. The formula is extended to some generalizations of the Levy Laplacian.

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References

  1. L. Accardi, P. Gibilisco, and I. V. Volovich, “Yang-Mills Gauge Fields as Harmonic Functions for the Levy Laplacians,” Russ. J. Math. Phys. 2(2), 235–250 (1994).

    MathSciNet  MATH  Google Scholar 

  2. L. Leandre and I. V. Volovich,“The Stochastic Levy Laplacian and Yang-Mills Equation on Manifolds,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4(2), 151–172 (2001).

    Article  MathSciNet  Google Scholar 

  3. B. O. Volkov, “Lévy Laplacian and the Gauge Fields,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15(4), 1250027 [19 pages] (2012).

    Article  Google Scholar 

  4. L. Accardi and O. G. Smolyanov, “Classical and Nonclassical Lévy Laplacians,” Dokl. Akad. Nauk 76(3), 801–805, (2007).

    MathSciNet  MATH  Google Scholar 

  5. L. Accardi and O. G. Smolyanov, “Generalized Lévy Laplacians and Cesaro Means,” Dokl. Akad. Nauk. 79(1), 90–93 (2009).

    MathSciNet  MATH  Google Scholar 

  6. L. Accardi and O. G. Smolyanov, ”Representations of Lévy Laplacians and Related Semigroups and Harmonic Functions,” Dokl. Akad. Nauk 65(3), 356–362 (2002).

    MathSciNet  MATH  Google Scholar 

  7. F. Gomez and O. G. Smolyanov, “Modified Lévy Laplacians,” Russ. J. Math. Phys. 15(1), 45–50 (2008).

    MathSciNet  MATH  Google Scholar 

  8. N. Obata, White Noise Calculus and Fock Space, Lect. Notes in Math. 1577 (Springer, Berlin, 1994).

    MATH  Google Scholar 

  9. L. Accardi, Y.-G. Lu, and I. V. Volovich, “Nonlinear Extensions of Classical and Quantum Stochastic Calculus and Essentially Infinite Dimensional Analysis,” in: Probability Towards 2000, Ed. by L. Accardi, C. C. Heyde, Lecture Notes in Statistics 128, pp. 1–33 (1998).

    Chapter  Google Scholar 

  10. H.-H. Kuo, White Noise Distribution Theory (CRC Press, Boca Raton, FL, 1996).

    MATH  Google Scholar 

  11. N. Obata, “Quadratic Quantum White Noises and Lévy Laplacian,” Nonlinear Anal. 47(4), 2437–2448 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Accardi, U. C. Ji, and K. Saitô, “Exotic Laplacians and Derivatives of White Noise,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14(1), 1–14 (2011).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    B. O. Volkov

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  1. B. O. Volkov
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Correspondence to B. O. Volkov.

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The research was supported by the Grant of the Government of the Russian Federation for the support of scientific researches of the Government of the Russian Federation in the Federal State Budget Educational Institution of Higher Professional Education “Lomonosov Moscow State University” according to the agreement no. 11.G34.31.0054.

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Volkov, B.O. Quantum probability and Levy Laplacians. Russ. J. Math. Phys. 20, 254–256 (2013). https://doi.org/10.1134/S1061920813020118

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