Prykhodko Y. Limit behaviour of local perturbations of Markov processes.

The thesis is devoted to the study of functional limit theorems for random processes with perturbations. A general method for obtaining functional limit theorems for locally perturbed processes is developed. The weak convergence of sequences of symmetric random walks with integrable local perturbations is proved and a complete classification of possible limit processes is performed. The limit behaviour of symmetric random walk with a reflecting barrier at zero is investigated for the case of non-integrable jumps from the barrier. The limit behaviour of sequences of solutions of stochastic differential equations with unbounded coefficients is obtained. The weak convergence of small random perturbations of stochastic differential equations with non-Lipschitz and degenerate coefficients is proved.