泛函随机微分方程是一类依赖于“过去”的随机动力系统，此类系统在生命、工程、系统科学等领域具有非常广泛的应用. 在过去的半个世纪，泛函随机微分方程在解过程以及“片段”过程的研究均得到了一定程度的发展. 然而，在解过程方面的研究成果要比“片段”过程丰富得多．相对于状态空间不依赖于“过去”的随机动力系统，泛函随机微分方程的“片段”过程长时间行为的研究还很不完善. 在本项目中, 我们将研究耗散型与非耗散型泛函随机微分方程(如布朗运动驱动的变时滞随机微分方程、中立型泛函随机微分方程、跳过程驱动的泛函随机微分方程等)不变测度的存在性、唯一性以及指数遍历性. 另外，我们还将研究遍历性在讨论(两时间尺度)泛函随机微分方程的平均值原理以及“片段”过程所生成马氏半群的超压缩性等问题中的应用.

Functional stochastic differential equations (SDEs) are stochastic dynamical systems, in which the state spaces depend on the past. Such equations have wide range of applications in life science, system science, engineering, etc. Over the past half-century, the study on solution processes and segment processes for functional SDEs has gained great development. However, the existing results on solution processes are more rich than those on segment processes. With contrast to stochastic dynamical systems without memory, the long-term behavior of functional SDEs is not yet complete. In this proposal, we shall investigate existence and uniqueness of invariant measures and exponential ergodicity for dissipative and non-dissipative functional SDEs, which cover SDEs with variable delays and neutral functional SDEs driven by Brwonian motions as well as functional SDEs subject to jump processes. Also, we shall study the applications of ergodicity in investigation on averaging principles and hypercontractivity of semigroups generated by segment processes for (two-time-scale) functional SDEs.