Article
Ufa Mathematical Journal
Volume 15, Number 2, pp. 55-64
Averaging of random affine transformations of variables in functions
Kalmetev R.Sh., Orlov Yu.N., Sakbaev V.Zh.
DOI:10.13108/2023-15-2-55
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We study the averaging of Feynmann-Chernov iterations of random operator-valued strongly continued functions, the values of which are bounded linear operators on separable Hilbert space. In this work we consider averaging for a certain system 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 variables in the functions. At the same time, the compositions of independent identically distributed random affine transformations are a non-commuting analogue of random walk.
For an operator-valued function being an averaging of Feynmann-Chernov iterations, we prove 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 Feynmann-Chernov iterations to the semigroup resolving the Cauchy problem for the corresponding Fokker-Planck equations.