Chernoff iterations as an averaging method for random affine transformations

For functions defined on a finite-dimensional vector space, we study compositions of their independent random affine transformations that represent a noncommutative analogue of random walks. Conditions on iterations of independent random affine transformations are established that are sufficient for convergence to a group solving the Cauchy problem for an evolution equation of shift along the averaged vector field and sufficient for convergence to a semigroup solving the Cauchy problem for the Fokker–Planck equation. Numerical estimates for the deviation of random iterations from solutions of the limit problem are presented. Initial-boundary value problems for differential equations describing the evolution of functionals of limit random processes are formulated and studied.