Averaging of random affine transformations of function arguments

We study compositions of independent random affine transformations of the argument of functions on a finite-dimensional linear space, which are a non-commutative analogue of random walks. Sufficient conditions are obtained for the convergence of the mathematical expectation of a sequence of Feynman-Chernov iterations of random affine transformations of a function argument to a semigroup solving the Cauchy problem for the corresponding inverse Kolmogorov equation.