随机动力系统中的回复性运动

Recurrence is one of the core topics in the study of dynamical systems. In this project, we mainly concern recurrent motions in stochastic dynamical systems or the impact of stochastic perturbations on recurrent motions. Precisely, we are to investigate the following problems: (1) we will establish the comparability principle of recurrent motions in general topological spaces and use it to study general recurrent motions of stochastic partial differential equations. (2) we will develop the stochastic averaging method and study almost periodic and almost automorphic motions for rapidly oscillating stochastic systems and stochastic fast-slow systems, and study further general recurrent motions of these systems. (3) we will study the stochastic reduction for space large-scale and time multi-scale systems, and investigate recurrent motions of these systems and the approximation relation between recurrent motions of original systems and that of the averaged stochastic systems. These topics need to combine the theory and methods of dynamical systems, stochastic analysis, ergodic theory and topology, and will deepen our understanding for the impact of stochastic perturbations on dynamics.

回复性是动力系统的核心研究课题之一，在此项目中我们主要关心随机动力系统中的回复性运动，或者说随机扰动对回复性运动的影响。具体地，我们拟研究如下几个问题：（1）建立一般拓扑空间上回复性运动的比较性原理，并利用它研究随机偏微分方程的一般回复性运动；（2）发展随机平均方法，研究时间快速振动的随机系统和随机快慢系统的概周期、几乎自守运动，进而研究一般的回复性运动；（3）研究大范围多尺度系统的随机降维，研究其回复性运动及降维之后的平均随机系统与原系统的回复解之间的逼近关系。这些研究内容需要综合运用动力系统、随机分析、遍历论和拓扑的相关理论与方法，并将使我们更深入理解随机扰动对动力学的影响。

项目执行期间，我们对如下问题开展了研究：我们研究了一般拓扑空间上回复性运动的比较性原理，并利用它研究了随机偏微分方程的一般回复性运动；研究了随机平均原理，特别是Bogolyubov第二型和全局平均原理，讨论了一般回复性运动的存在唯一性和全局渐近稳定性等。我们证明了McKean-Vlasov型随机微分方程在Lyapunov条件下的存在唯一性和指数遍历性。我们证明了线性随机微分方程的Lyapunov指数在充分小扰动之下符号可以发生改变，但不能所有Lyapunov指数的符号同时发生改变。