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

Weak convergence of Markovian random evolution in a multidimensional space