Stochastic Processes on the Group of Orthogonal Matrices and Evolution Equations Describing Them

Abstract

Stochastic processes that take values in the group of orthogonal transformations of a finite-dimensional Euclidean space and are noncommutative analogues of processes with independent increments are considered. Such processes are defined as limits of noncommutative analogues of random walks in the group of orthogonal transformations. These random walks are compositions of independent random orthogonal transformations of Euclidean space. In particular, noncommutative analogues of diffusion processes with values in the group of orthogonal transformations are defined in this manner. Kolmogorov backward equations are derived for these processes.

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REFERENCES

  1. 1

    I. Ya. Aref’eva and I. V. Volovich, “Quasi-averages in random matrix models,” Proc. Steklov Inst. Math. 306, 1–8 (2019).

    MathSciNet  Article  Google Scholar 

  2. 2

    Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Unbounded random operators and Feynman formulas,” Izv. Math. 80 (6), 1131–1158 (2016).

    MathSciNet  Article  Google Scholar 

  3. 3

    I. V. Volovich and V. Zh. Sakbaev, “On quantum dynamics on C*-algebras,” Proc. Steklov Inst. Math. 301, 25–38 (2018).

    MathSciNet  Article  Google Scholar 

  4. 4

    Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, “Randomized Hamiltonian mechanics,” Dokl. Math. 99 (3), 313–316 (2019).

    Article  Google Scholar 

  5. 5

    V. Zh. Sakbaev, “Averaging of random flows of linear and nonlinear maps,” J. Phys. Conf. Ser. 990, 012012 (2018).

    MathSciNet  Article  Google Scholar 

  6. 6

    E. B. Dynkin, Theory of Markov Processes (Fizmatgiz, Moscow, 1963; Dover, New York, 2006).

  7. 7

    T. M. Liggett, Interacting Particle Systems (Springer, New York, 2006).

    Google Scholar 

  8. 8

    V. Zh. Sakbaev, O. G. Smolyanov, and N. N. Shamarov, “Non-Gaussian Lagrangian Feynman–Kac formulas,” Dokl. Math. 90 (1), 416–418 (2014).

    MathSciNet  Article  Google Scholar 

  9. 9

    Yu. N. Orlov, V. Zh. Sakbaev, and D. V. Zavadsky, “Operator random walks and quantum oscillator,” Lobachevskii J. Math. 41 (4), 676–685 (2020).

    MathSciNet  Article  Google Scholar 

  10. 10

    M. Loève, Probability Theory (Springer-Verlag, New York, 1977).

    Google Scholar 

  11. 11

    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  Article  Google Scholar 

  12. 12

    M. C. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis (Academic, New York, 1972).

  13. 13

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

    MathSciNet  Article  Google Scholar 

  14. 14

    T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).

    Google Scholar 

  15. 15

    L. N. Slobodetskii, “Generalized Sobolev spaces and their applications to boundary value problems for partial differential equations,” Uchen. Zap. Leningr. Gos. Ped. Inst. 197, 54–112 (1958).

    Google Scholar 

  16. 16

    A. A. Borovkov, Probability Theory (Fizmatlit, Moscow, 1986; Springer, London, 2013).

Download references

Funding

This work was performed at the Laboratory of Infinite-Dimensional Analysis and Mathematical Physics (headed by Smolyanov) of the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University.

Smolyanov acknowledges the support of the Scientific Program “Fundamental Problems in Mechanics and Mathematics” of the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University.

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Correspondence to K. Yu. Zamana or V. Zh. Sakbaev or O. G. Smolyanov.

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Translated by I. Ruzanova

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Zamana, K.Y., Sakbaev, V.Z. & Smolyanov, O.G. Stochastic Processes on the Group of Orthogonal Matrices and Evolution Equations Describing Them. Comput. Math. and Math. Phys. 60, 1686–1700 (2020). https://doi.org/10.1134/S0965542520100140

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Keywords:

  • random linear operator
  • random operator-valued function
  • operator-valued stochastic process
  • law of large numbers
  • Kolmogorov equation