Averaging of random affine transformations of functions domain

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.