Weak convergence of Markovian random evolution in a multidimensional space

TL;DR

This paper proves that certain Markovian random evolutions in multidimensional space converge weakly to Wiener and diffusion processes, using singular perturbation techniques and establishing relative compactness.

Contribution

It demonstrates the weak convergence of both symmetric and non-symmetric Markovian random evolutions to continuous processes in multidimensional space, extending previous results.

Findings

Symmetric random evolutions converge to Wiener process.

Non-symmetric evolutions converge to diffusion with drift.

Relative compactness of evolution families is established.

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.