随机泛函微分方程的动力学性态

As a former part, this project establishes the fundamental theory of random functional differential equations (RFDE, for short), called real noise case, including the existence and uniqueness of solutions, continuous dependence and differentiability with respect to initial value functions, and measurability. This leads us to prove that the solutions for RDE generate a random dynamical systems (RDS) on continuous functions space if every solution can be extended to positive infinity, a sufficient condition to guarantee such an extension is that the right-hand functions either have global Lipschitz property or satisfy linear growth conditions. A criterion will be provided for a closed subset of continuous functions space, and sufficient conditions for RDS to be dissipative and possess a random attractor will be given. The comparison principle for RFDE will be proved, which leads us to set up the theory of order-preserving, or eventually strong order-preserving RDS generated by RFDE. As a consequece, a sub-stationary solution or upper-stationary solution with compact orbit closure will converge to a stationary solution, which is a stationary process, and has certain stability in dynamics, and every compact random attractor contains such a stationary solution. Under the assumption that right-hand functions are sublinear, we will show that RFDE generates a sublinear RDS by comparison principle. Then we can discuss the classification of limit sets for pull-back trajectories of such a sublinear RDS and provide many applications. Based on the former part, we will investigate stochastic functional differential equations (SFDE). The key point here is to find the conditions of diffusion term on the conjugacy of stochastic and random functional differential equations, and at least we will construct the conjugacy if diffusion term with linear growth is either nonzero or only takes zero value at the origin with nonzero derivatives. Then such kind of SFDEs generate RDS. The results in former part can help us to form the conditions for the solutions of SFDEs generate order-preserving RDS、eventualy strongly order-preserving RDS and sublinear RDS. The random attractor 、 stastinary solution 、ergodicity and dichotomy and trichotomy of limit sets will be extensively explored.

该项目首先建立带Real Noise的随机泛函微分方程（RFDE）的基本理论，包括解的存在唯一性、对初值函数的连续性和可微性等。证明解函数的三元可测性，提供条件保证RFDE以连续函数为相空间生成随机动力系统（RDS）。提供相空间的闭子集为正不变的准则。给出这一RDS耗散和吸引子存在的充分条件。提供这类随机泛函微分方程的比较原理和随机动力系统保持相空间序的条件，进而得到序保持随机动力系统。通过给出变分方程的不可约性定义和判定，得到产生最终强序保持随机动力系统的准则，由此得到随机吸引子含有具有一定稳定意义的平稳解。在右端函数次线性的条件下，讨论拉回轨线极限集的分类。接着，研究由Brown运动驱动的随机泛函微分方程(SFDE)，寻求扩散项的条件，使得SFDE与某些RFDE可测共轭，由此获得其解生成RDS、强序保持、次线性的条件，并获得吸引子、平稳解、以及极限集的二分性和三分性等结果。

奠定了Real Noise随机微分方程的基本理论: 在局部有界和Lipschtz的假设之下，系统初值问题存在唯一不可延伸的解， 它关于时间和初值是连续的、关于样本点是可测的；当系统关于第二变量是连续可微时，解关于初值是连续可微的，其导算子满足对应的变分方程; 当系统具有整体Lipschitz条件， 或被线性控制， 或拥有某些性质的Liapunov函数时， 其解生成随机动力系统（RDS）；提供了相空间的闭子集为正不变的准则，给出这一RDS耗散和吸引子存在的充分条件；拟单调系统的解生成随机单调动力系统、证明可测版本的Riesz 表示定理，并导出不可约条件，推出拟单调和不可约条件生成随机强单调动力系统，由此得到随机吸引子含有稳定的平稳解； 证明次线性系统的拉回轨道具有二分性； 证明布朗运动驱动的两类（Retarded, 中立）泛函微分方程的比较原理；证明加法噪声驱动的反馈系统具有全局稳定的随机奇点；建立新的竞争映射负载单形存在性， 在一定意义下，分类三维Leslie/Gower模型和Atkinson-Allen模型的动力学性态。