On limit measures and their supports for stochastic ordinary differential equations
On limit measures and their supports for stochastic ordinary differential equations
复制标题
DOI：
10.1016/j.jde.2023.04.005
复制
发表时间：
2022-01
影响因子：
2.4
通讯作者：
Tianyuan Xu;Lifeng Chen;Jifa Jiang
中科院分区：
文献类型：
--
作者：
Tianyuan Xu;Lifeng Chen;Jifa Jiang
本文研究欧氏空间上随机常微分方程平稳测度的极限测度，并试图确定未扰动系统的哪些不变测度将继续存在。在弱紧性条件下，在允许Freidlin-Wentzell或Dembo-Zeitouni大偏差原理的假设下，证明了极限测度集中在远离排斥子的地方，这些排斥子是拓扑传递的，或等价类，或允许Lebesgue测度为零.我们也排除了非循环鞍链或陷阱链上极限测度的集中。这说明极限测度集中在李雅普诺夫稳定紧不变集上。应用于Morse-Smale系统、公理A系统（包括结构稳定系统和分离星星系统）、梯度或类梯度系统、具有Poincaré-Bendixson性质的有限个极限集的系统，得到了极限测度存在于Liapunov稳定临界元、Liapunov稳定基本集、Liapunov稳定平衡点和Liapunov稳定极限集（包括平衡点）上，极限环和鞍环或陷阱环。给出了四个具有唯一极限测度的非平凡例子，其中两个例子具有无穷等价类且其支撑为Liapunov稳定周期轨道，一个例子支撑在鞍点上.
This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin–Wentzell or Dembo–Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This illustrates that limit measures are concentrated on Liapunov stable compact invariant sets. Applications are made to the Morse–Smale systems, the Axiom A systems including structural stability systems and separated star systems, the gradient or gradient-like systems, those systems possessing the Poincaré–Bendixson property with a finite number of limit sets to obtain that limit measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria and Liapunov stable limit sets including equilibria, limit cycles and saddle or trap cycles, respectively. Four nontrivial examples admitting a unique limit measure are provided, two of which possess infinite equivalent classes and their supports are Liapunov stable periodic orbits, one is supported on saddle-nodes.