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、生物和工程等领域有着广泛的应用。其中许多问题涉及随机扰动动力系统在扰动强度无限增长的情况下的渐近行为。这项研究将研究这类系统的概率有界性、遍历性、概率1有界性和稳定性。这将允许研究者在一般空间和扰动类型中发展系统的随机平均原理，并建立关于大时间扰动系统的分布弱收敛到扩散马尔可夫过程的分布的定理。本文的研究方法是基于鞅理论、复杂系统的随机微分方程组和随机微分方程组的极限定理。研究人员将考虑扩展相空间中跳跃过程的新的一般极限定理，以研究复杂的马尔可夫过程到更简单的马尔可夫过程的弱收敛。具有随机扰动的动力系统在力学、物理、生物和工程等领域有着广泛的应用。研究随机扰动动力系统在扰动强度无界增长条件下的渐近行为有许多未解决的问题。这项研究将调查这类系统的主要性质。这项研究将促进对随机摄动系统行为的认识，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.