Compositions of independent random operators and related differential equations

Abstract

Iterations of independent random linear operators in the Hilbert space of square integrable functions on a finite dimensional Euclidean space are studied. Random operator under consideration take values in the algebra of operators which is generated by an operators of a shift on a vector of Euclidean space of the argument of a function or the argument of its Fourier image, operators of orthogonal mapping and operators of contraction of argument space. We obtain the conditions sufficient to convergence of a sequence of mean values of compositions of operator valued processes with values in the considered algebra of linear operators to the semigroup describing the diffusion in finite dimensional Euclidean space. Generators of limit semigroups are described.