Dynamical systems with random perturbations have been used intensively in mechanics, physics, biology, and engineering. Many of these problems concern the asymptotic behavior of randomly perturbed dynamical systems under unbounded growth of intensities of perturbation. This research will investigate boundedness in probability, ergodicity, boundedness with probability 1, and stability for such systems. This will allow the investigator to develop the principle of random averaging for systems in general spaces and types of perturbations and to establish theorems on weak convergence of the distributions of a perturbed system for large times to distributions of a diffusion Markov process. The methods of investigation are based on martingale theory, stochastic differential equations for complex systems, and limit theorems for stochastic differential equations. The investigator will consider new general limit theorems for jump processes in extending phase spaces to study the weak convergence of complex Markov processes to more simple ones. Dynamical systems with random perturbations have been used intensively in mechanics, physics, biology, and engineering. There are many unsolved problems which are connected with the study of the asymptotic behavior of randomly perturbed dynamical systems under conditions of unbounded growth on the intensities of perturbation. This research will investigate the main properties of such systems. The research will advance the knowledge of the behavior of stochastically perturbed systems and will provide significant applications to the engineering sciences.
