We first introduce the concept of weak random periodic solutions of random dynamical systems. Then, we discuss the existence of such periodic solutions. Further, we introduce the definition of weak random periodic measures and study their relationship with weak random periodic solutions. In particular, we establish the existence of invariant measures of random dynamical systems by virtue of their weak random periodic solutions. We use concrete examples to illustrate the weak random periodic phenomena of dynamical systems induced by random and stochastic differential equations.

中文翻译：

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随机动力系统的弱随机周期解

我们首先介绍随机动力系统弱随机周期解的概念。然后，我们讨论这种周期解的存在性。此外，我们介绍了弱随机周期测度的定义并研究了它们与弱随机周期解的关系。特别是，我们利用随机动力系统的弱随机周期解来建立随机动力系统不变测度的存在性。我们用具体的例子来说明由随机和随机微分方程引起的动力系统的弱随机周期现象。