Mathematical Sciences: Asymptotical Analysis of Dynamical Systems with Markov and Semi Markov Random Perturbations

具有随机扰动的动力系统在力学、物理学、生物学和工程学中有着广泛的应用。 这些问题中有许多涉及到随机扰动动力系统在扰动强度无限增长下的渐近行为。 本研究将探讨此类系统的概率有界性、遍历性、概率为1的有界性和稳定性。 这将使调查员制定原则的随机平均系统在一般的空间和类型的扰动和建立定理弱收敛的分布的扰动系统的大时间的分布的扩散马尔可夫过程。 调查的方法是基于鞅理论，复杂系统的随机微分方程，随机微分方程的极限定理。 研究者将考虑扩展相空间中跳跃过程的新的一般极限定理，以研究复杂马尔可夫过程向更简单过程的弱收敛。 具有随机扰动的动力系统在力学、物理学、生物学和工程学中有着广泛的应用。 在扰动强度无限增长的条件下，研究随机扰动动力系统的渐近行为有许多未解决的问题。 本研究将探讨这类系统的主要性质。 该研究将推进对随机扰动系统行为的认识，并将为工程科学提供重要的应用。

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.