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Unitary representation of walks along random vector fields and the Kolmogorov–Fokker–Planck equation in a Hilbert space

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Abstract

Random Hamiltonian flows in an infinite-dimensional phase space is represented by random unitary groups in a Hilbert space. For this, the phase space is equipped with a measure that is invariant under a group of symplectomorphisms. The obtained representation of random flows allows applying the Chernoff averaging technique to random processes with values in the group of nonlinear operators. The properties of random unitary groups and the limit distribution for their compositions are described.

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References

  1. V. Zh. Sakbaev, “Flows in infinite-dimensional phase space equipped with a finitely-additive invariant measure,” Mathematics, 11, 1161, 49 pp. (2023).

    Google Scholar 

  2. V. M. Busovikov and V. Zh. Sakbaev, “Invariant measures for Hamiltonian flows and diffusion in infinitely dimensional phase spaces,” Internat. J. Modern Phys. A, 37, 2243018, 15 pp. (2022).

    ADS  MathSciNet  CAS  Google Scholar 

  3. V. A. Glazatov and V. Zh. Sakbaev, “Measures on Hilbert space invariant with respect to Hamiltonian flows,” Ufa Math. J., 14, 3–21 (2022).

    MathSciNet  Google Scholar 

  4. N. N. Vakhania, V.I. Tarieladze, and S. A. Chobanyan, Probability Distributions on Banach Spaces (Mathematics and its Applications, Vol. 14), Springer, Dordrecht (1987).

    Google Scholar 

  5. R. L. Baker, “ ‘Lebesgue measure’ on \(R^{\infty}\). II,” Proc. Amer. Math. Soc., 132, 2577–2591 (2004).

    MathSciNet  Google Scholar 

  6. V. V. Kozlov and O. G. Smolyanov, “Hamiltonian approach to secondary quantization,” Dokl. Math., 98, 571–574 (2018).

    MathSciNet  Google Scholar 

  7. O. G. Smolyanov and N. N. Shamarov, “Schrödinger quantization of infinite-dimensional Hamiltonian systems with a nonquadratic Hamiltonian function,” Dokl. Math., 101, 227–230 (2020).

    MathSciNet  Google Scholar 

  8. I. V. Volovich, “Complete integrability of quantum and classical dynamical systems,” \(p\)-Adic Numbers, Ultrametric Analysis and Applications, 11, 328–334 (2019).

    MathSciNet  Google Scholar 

  9. J. Gough, Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, “Random quantization of Hamiltonian systems,” Dokl. Math., 103, 122–126 (2021).

    MathSciNet  Google Scholar 

  10. V. Zh. Sakbaev, “Averaging of random walks and shift-invariant measures on a Hilbert space,” Theoret. and Math. Phys., 191, 886–909 (2017).

    ADS  MathSciNet  Google Scholar 

  11. V. Zh. Sakbaev and O. G. Smolyanov, “Lebesgue–Feynman measures on infinite dimensional spaces,” Internat. J. Theoret. Phys., 60, 546–550 (2021).

    ADS  MathSciNet  Google Scholar 

  12. T. Gill, A. Kirtadze, G. Pantsulaia, and A. Plichko, “Existence and uniqueness of translation invariant measures in separable Banach spaces,” Funct. Approx. Comment. Math., 50, 401–419 (2014).

    MathSciNet  Google Scholar 

  13. D. V. Zavadsky, “Shift-invariant measures on sequence spaces,” Proceedings of MIPT, 9, 142–148 (2017).

    Google Scholar 

  14. P. R. Chernoff, “Note on product formulas for operator semigroups,” J. Funct. Anal., 2, 238–242 (1968).

    MathSciNet  Google Scholar 

  15. Yu. N. Orlov, V. Zh. Sakbaev, and E. V. Shmidt, “Operator approach to weak convergence of measures and limit theorems for random operators,” Lobachevskii J. Math., 42, 2413–2426 (2021).

    MathSciNet  Google Scholar 

  16. V. I. Arnold, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, Vol. 60), Springer, New York–Heidelberg (1978).

    Google Scholar 

  17. A. Yu. Khrennikov, “Symplectic geometry on an infinite-dimensional phase space and an asymptotic representation of quantum averages by Gaussian functional integrals,” Izv. Math., 72, 127–148 (2008).

    MathSciNet  Google Scholar 

  18. V. Zh. Sakbaev, “Random walks and measures on Hilbert space that are invariant with respect to shifts and rotations,” J. Math. Sci., 241, 469–500 (2019).

    Google Scholar 

  19. D. V. Zavadsky and V. Zh. Sakbaev, “Diffusion on a Hilbert space equipped with a shift- and rotation-invariant measure,” Proc. Steklov Inst. Math., 306, 102–119 (2019).

    MathSciNet  Google Scholar 

  20. V. M. Busovikov and V. Zh. Sakbaev, “Sobolev spaces of functions on a Hilbert space endowed with a translation-invariant measure and approximations of semigroups,” Izv. Math., 84, 694–721 (2020).

    MathSciNet  Google Scholar 

  21. T. Kato, Perturbation Theory for Linear Operators (Classics in Mathematics, Vol. 132), Springer, Berlin (1995).

    Google Scholar 

  22. M. Reed and B. Simon, Methods of modern mathematical physics, Vol. I: Functional analysis, Academic Press, New York–London (1972).

    Google Scholar 

  23. Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, “Feynman formulas and the law of large numbers for random one-parameter semigroups,” Proc. Steklov Inst. Math., 306, 196–211 (2019).

    MathSciNet  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation under grant No. 19-11-00320, https://rscf.ru/en/project/19-11-00320/.

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Authors and Affiliations

  1. Phystech School of Applied Mathematics and Informatics, Moscow Institute for Physics and Technology (National Research University), Dolgoprudny, Moscow Region, Russia

    V. M. Busovikov

  2. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    V. M. Busovikov

  3. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

    Yu. N. Orlov & V. Zh. Sakbaev

Authors
  1. V. M. Busovikov
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  2. Yu. N. Orlov
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  3. V. Zh. Sakbaev
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Correspondence to V. Zh. Sakbaev.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 218, pp. 238–257 https://doi.org/10.4213/tmf10548.

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Busovikov, V.M., Orlov, Y.N. & Sakbaev, V.Z. Unitary representation of walks along random vector fields and the Kolmogorov–Fokker–Planck equation in a Hilbert space. Theor Math Phys 218, 205–221 (2024). https://doi.org/10.1134/S004057792402003X

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  • DOI: https://doi.org/10.1134/S004057792402003X

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