V.Zh. Sakbaev^a, O.G. Smolyanov^b,a

^aMoscow Institute of Physics and Technology, Dolgoprudny, 141701 Russia

^bLomonosov Moscow State University, Moscow, 119991 Russia

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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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2. Bogachev V.I., Smolyanov O.G. Topological Vector Spaces and Their Applications. Springer, 2017. x, 456 p. doi: 10.1007/978-3-319-57117-1.

3. Sakbaev V.Zh. On the law of large numbers for compositions of independent random semigroups.  Russ. Math., 2016, vol. 60, no. 10, pp. 72–76. doi: 10.3103/S1066369X16100121.  (in Russian).

4. Bogachev V.I. Measure Theory. Vol. 1. Berlin, Heidelberg, Springer, 2007. xviii+500 p.

Recieved

November 14, 2017

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

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