Feynman calculus for random operator-valued functions and their applications
V.Zh. Sakbaeva, O.G. Smolyanovb,a
aMoscow Institute of Physics and Technology, Dolgoprudny, 141701 Russia
bLomonosov Moscow State University, Moscow, 119991 Russia
For citation: Sakbaev V.Zh., Smolyanov O.G. Feynman calculus for random operator-valued functions and their applications. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 373–383.
Для цитирования: Sakbaev V.Zh., Smolyanov O.G. Feynman calculus for random operator-valued functions and their applications // Учен. зап. Казан. ун-та. Сер. Физ.- матем. науки. – 2018. – Т. 160, кн. 2. – С. 373–383.
Abstract
The Feynman–Chernoff iteration of a random semigroup of bounded linear operators in the Hilbert space has been considered. The convergence of mean values of the Feynman–Chernoff iteration of a random semigroup has been studied. The estimates of the deviation of compositions of the independent identically distributed random semigroup from its mean value have been obtained as the large numbers law for the sequence of compositions of the independent random semigroup has been investigated. The relationship between the semigroup properties of the mean values of the random operator-valued function and the property of independence of the increments of the random operator-valued function has been analyzed. The property of asymptotic independence of the increments of the Feynman–Chernoff iteration of the random semigroup has been discussed. The independization of the random operator-valued function has been defined as the map of this random operator function into the sequence of random operator-valued functions, which has asymptotically independent increments. The examples of independization (which is similar to the Feynman–Chernoff iteration) of the random operator-valued function have been given.
Keywords: random operator, random semigroup, Feynman–Chernoff iteration, large numbers law
Acknowledgements. The work was performed according to the Russian Government Program of Competitive Growth of Moscow Institute of Physics and Technology (project 5-100).
References
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Recieved
November 14, 2017
Sakbaev Vsevolod Zhanovich, Doctor of Physics and Mathematics, Professor of the Higher Mathematics Department
Moscow Institute of Physics and Technology
Institutskiy per., 9, Dolgoprudny, Moscow Region, 141701 Russia
E-mail: fumi2003@mail.ru
Smolyanov Oleg Georgievich, Doctor of Physics and Mathematics, Professor of the Department of Theory of Functions and Functional Analysis, Faculty of Mechanics and Mathematics, Moscow State University; Professor of the Department of Higher Mathematics
Lomonosov Moscow State University
Leninskie Gory, 1, Moscow, 119991 Russia
Moscow Institute of Physics and Technology
Institutskiy per., 9, Dolgoprudny, Moscow Region, 141701 Russia
E-mail: smolyanov@yandex.ru
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