Shvets O. Random evolutions with equilibrium on increasing time intervals

The thesis is devoted to the research of random evolution and stochastic approximation procedure in Markov environment under singular, diffusive and impulsive perturbations in average schema, diffusion approximation schema and asymptotic small diffusion schema in case of balance condition (with equilibrium). It has been established sufficient conditions for the convergence of a multi-dimensional stochastic process in the case of dependence of the regression function on the environment, which is described by Markov switching in average schema. There is obtained sufficient conditions for the convergence of a stochastic approximation procedure in the case of diffusive and singular perturbation on the stochastic differential equation in diffusion approximation schema. In this work it has been prooved the convergence of exponential generator for random evolution with singular perturbation and asymptotic small diffusion. Also it has been obtained the sufficient conditions for the convergence of a stochastic approximation procedure in asymptotic small diffusion schema and in diffusion approximation schema with impulse perturbation, which is described by processes with independent increase in case of local balance condition. Keywords: random evolution, Markov process, small parameter, singular perturbation problem, exponential generator, stochastic approximation procedure, Lyapunov function.