Weak convergence of Markovian random evolution in a multidimensional space

Abstract

We study Markovian symmetry and non-symmetry random evolutions in ${\mathbf{R}}^n$. Weak convergence of Markovian symmetry random evolution to Wiener process and of Markovian non-symmetry random evolution to a diffusion process with drift is proved using problems of singular perturbation for the generators of evolutions. Relative compactness in ${\mathbf{D}}_{{\mathbf{R}}^n\times \Theta }[0,\infty )$ of the families of Markovian random evolutions is also shown.