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
I. Ya. Aref’eva and I. V. Volovich, “Quasi-averages in random matrix models,” Proc. Steklov Inst. Math. 306, 1–8 (2019).
Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Unbounded random operators and Feynman formulas,” Izv. Math. 80 (6), 1131–1158 (2016).
I. V. Volovich and V. Zh. Sakbaev, “On quantum dynamics on C*-algebras,” Proc. Steklov Inst. Math. 301, 25–38 (2018).
Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, “Randomized Hamiltonian mechanics,” Dokl. Math. 99 (3), 313–316 (2019).
V. Zh. Sakbaev, “Averaging of random flows of linear and nonlinear maps,” J. Phys. Conf. Ser. 990, 012012 (2018).
E. B. Dynkin, Theory of Markov Processes (Fizmatgiz, Moscow, 1963; Dover, New York, 2006).
T. M. Liggett, Interacting Particle Systems (Springer, New York, 2006).
V. Zh. Sakbaev, O. G. Smolyanov, and N. N. Shamarov, “Non-Gaussian Lagrangian Feynman–Kac formulas,” Dokl. Math. 90 (1), 416–418 (2014).
Yu. N. Orlov, V. Zh. Sakbaev, and D. V. Zavadsky, “Operator random walks and quantum oscillator,” Lobachevskii J. Math. 41 (4), 676–685 (2020).
M. Loève, Probability Theory (Springer-Verlag, New York, 1977).
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).
M. C. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis (Academic, New York, 1972).
P. Chernoff, “Note on product formulas for operator semigroups,” J. Funct. Anal. 2 (2), 238–242 (1968).
T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).
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).
A. A. Borovkov, Probability Theory (Fizmatlit, Moscow, 1986; Springer, London, 2013).
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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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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DOI: https://doi.org/10.1134/S0965542520100140