随机泛函微分方程的平稳性和稳定性

在本项目中，我们主要研究随机泛函微分方程的平稳性和稳定性，特别关注噪声形式以及反馈函数的变化对于随机系统平稳性和稳定性的影响。首先，对于一些随机集的可测性，给出更加细致的刻画。其次，发展单调随机动力系统的理论和方法，研究反馈函数为单调或反单调时随机系统的平稳性和稳定性，建立一般性的理论框架：通过构造合适的算子并且利用Banach压缩映像原理来定性地保证非平凡平稳解的存在性，进而证明此平稳解在拉回轨道的意义下是全局吸引的。接着，利用第二部分的理论结果继续讨论反馈函数是非单调的情形，利用比较原理和上下解技术，仔细分析极限集的结构，证明其就是一个单点集，从而得到随机系统的平稳性和稳定性。最后，研究由乘法噪声驱动的拟单调次线性随机系统极限集的三分性和两分性，并且对于具体的随机泛函微分方程，提供适当的充分性条件，验证拉回轨道极限集的三分性和两分性。

In this project, we mainly consider the stationarity and stability of stochastic functional differential equations. Precisely, we aim to concern the influence of the noise and the feedback functions on the stationarity and stability of stochastic systems. First of all, we will establish some more detailed measurability of the random sets. Secondly, in the case that feedback functions are monotone or anti-monotone, using and developing the theory and the methods of monotone random dynamical systems, we will consider the stationarity and stability of stochastic systems and give the general framework. The main thought is to construct some operators and use the Banach fixed-point theorem to qualitatively guarantee the existence of nontrivial stationary solutions, and then we need to prove that this stationary solution is globally attractive in the sense of the pull-back trajectories. Next, taking advantage of the results of part two to discuss the case that feedback functions are non-monotone, we will use the comparison principle and the techniques of upper and lower solutions to carefully analyze the structure of limit sets, and prove that the limit set actually is one single point. This fact will lead to the stationarity and stability of stochastic systems. Finally, we will go into the limit set trichotomy theorem for sublinear random dynamical systems driven by the multiplicative noise. For specific stochastic functional differential equations, we will give some sufficient conditions to verify the limit set trichotomy theorem for the pull-back trajectories.