平面随机微分方程依分布周期解的研究

The periodic phenomena generally exist in nature, such as the celestial movement, the cell division of organisms, the vibration of waves, the change of four seasons, and so on. Periodic solutions of differential equations is one of the basic tools depicting the above periodic phenomena. However, it is worth noting that the essence of the world is random and any periodic phenomena are always disturbed by the noise in the environment. Therefore, it has important theoretical value and practical significance to investigate the existence of periodic solutions of stochastic differential equations. In this project, we explore the existence of periodic solutions in distribution of stochastic differential equations on the plane. We establish sufficient criterion for the existence of periodic solutions in distribution of linear and nonlinear stochastic differential equations on the plane. In particular, we focus on investigating the relationship between the existence of periodic solutions in distribution and the bounded solutions of stochastic differential equations, and give the Massera criterion of stochastic differential equations. Therefore, this project may reveal the intrinsic effect and essential characteristics of the environmental noise for the periodic phenomena.

周期现象广泛存在于自然界中，如天体运动、生物体的细胞分裂、波震动、四季的气候变化等。微分方程周期解是刻画上述周期现象的基本工具之一。然而值得注意的是世界的本质是随机的，任何周期现象都会受到环境噪声的影响，因而研究随机微分方程周期解能够更好地描述自然界中的周期现象，更具理论价值和现实意义。本项目拟研究平面随机微分方程依分布周期解的存在性。建立平面上线性和非线性随机微分方程依分布周期解存在的判别准则。着重分析平面随机微分方程依分布周期解的存在性与有界解之间的联系，进而建立平面随机微分方程的Massera准则。因而本项目的研究有助于揭示环境噪声对周期现象的内在影响及其本质特征。

周期现象广泛存在于自然界中，如天体运动、生物体的细胞分裂、波震动、四季的气候变 化等。微分方程周期解是刻画上述周期现象的基本工具之一。然而值得注意的是世界的本质是 随机的，任何周期现象都会受到环境噪声的影响，因而研究随机微分方程周期解能够更好地描 述自然界中的周期现象，更具理论价值和现实意义。本项目研究了一些随机模型的动力学行为，尤其是平面上线性和非线性随机微分方程依分布周期解的存在性。本项目的研究在一定意义下揭示了环境噪声对周期现象的内在影响及其本质特征。本项目通过三年的研究工作，已基本上完成项目计划内容。在本项目的执行过程中，共发表被SCI收录的学术论文8篇，另有1篇接收，另外在项目资助下出版专著1部。