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Ufa Mathematical Journal, 2023, Volume 15, Issue 2, Pages 55–64
DOI: https://doi.org/10.13108/2023-15-2-55
(Mi ufa653)
 

Averaging of random affine transformations of functions domain

R. Sh. Kalmetevab, Yu. N. Orlova, V. Zh. Sakbaevabc

a Keldysh Institute of Applied Mathematics of Russian Academy of Sciences
b Miusskaya sq. 4, 125047, Moscow, Russia
c Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevsky str. 112, 450077, Ufa, Russia
References:
Abstract: We study the averaging of Feynman-Chernoff iterations of random operator-valued strongly continuous functions, the values of which are bounded linear operators on separable Hilbert space. In this work we consider averaging for a certain family of such random operator-valued functions. Linear operators, being the values of the considered functions, act in the Hilbert space of square integrable functions on a finite-dimensional Euclidean space and they are defined by random affine transformations of the functions domain. At the same time, the compositions of independent identically distributed random affine transformations are a non-commutative analogue of random walk.
For an operator-valued function being an averaging of Feynman-Chernoff iterations, we prove an upper bound for its norm and we also establish that the closure of the derivative of this operator-valued function at zero is a generator a strongly continuous semigroup. In the work we obtain sufficient conditions for the convergence of the mathematical expectation of the sequence of Feynman-Chernoff iterations to the semigroup resolving the Cauchy problem for the corresponding Fokker-Planck equation.
Keywords: Feynman-Chernoff iterations, Chernoff theorem, operator-valued random process, Fokker-Planck equation.
Received: 21.12.2022
Russian version:
Ufimskii Matematicheskii Zhurnal, 2023, Volume 15, Issue 2, Pages 55–64
Document Type: Article
UDC: 517.983
Language: English
Original paper language: Russian
Citation: R. Sh. Kalmetev, Yu. N. Orlov, V. Zh. Sakbaev, “Averaging of random affine transformations of functions domain”, Ufimsk. Mat. Zh., 15:2 (2023), 55–64; Ufa Math. J., 15:2 (2023), 55–64
Citation in format AMSBIB
\Bibitem{KalOrlSak23}
\by R.~Sh.~Kalmetev, Yu.~N.~Orlov, V.~Zh.~Sakbaev
\paper Averaging of random affine transformations of functions domain
\jour Ufimsk. Mat. Zh.
\yr 2023
\vol 15
\issue 2
\pages 55--64
\mathnet{http://mi.mathnet.ru/ufa653}
\transl
\jour Ufa Math. J.
\yr 2023
\vol 15
\issue 2
\pages 55--64
\crossref{https://doi.org/10.13108/2023-15-2-55}
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  • https://doi.org/10.13108/2023-15-2-55
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