现实中的现象往往是多个系统与随机环境相互影响的结果。无界区域上的随机偏微分方程理论是随机分析理论中的重要组成部分，方法和技术有待丰富。本项目将研究有界和无界区域上的一类趋化-流体动力学中的随机偏微分方程。该模型是耦合方程组，具有复杂的非线性项，随机噪声的引入将使该模型产生有别于现有随机偏微分方程长时间行为一般理论的行为。这些使得研究本项目的方法和技术需要确定型模型的研究方法和技术与随机分析理论的结合和创新，特别是对于该模型长时间行为的研究。本项目的研究将对随机噪声驱动的耦合方程组的研究和无界区域上随机偏微分方程的研究提供可以借鉴的方法，对随机偏微分方程理论研究本身有重要的理论意义。趋化现象是生物学中的普遍现象，该现象的研究具有很强的应用背景。随机噪声的引入将使趋化-流体模型产生新的长时间行为，该行为的研究将对趋化现象的理解有重要的理论意义，以及潜在的应用价值。

Real-world phenomena always contain several coupled systems and are influenced by random environments. In recent years, the research on coupled systems in random environments has attracted more and more attention. Stochastic partial differential equations in unbounded domains are an important part in Stochastic Analysis, but there are not many methods and techniques to deal with the problems on this topic. This project aims to study a class of Stochastic Partial Differential Equations in Chemotaxis-Fluid Dynamics on bounded and unbounded domains. These models are coupled systems and have involved nonlinear terms. Introducing stochastic noises will lead to new long-time behaviors which is different from the general theories of stochastic partial differential equations. These lead to that it is highly non-trivial to deal with the stochastic cases, especially to study long-time behaviors. It is not enough to simply combine the methods and techniques used in the study of deterministic models and the theories of stochastic analysis. New ideas and techniques should been raised. Our project will have a certain reference value in studying such coupled systems in random environment and stochastic partial differential equations in unbounded domains, and it is of vital theoretical significance in studying stochastic partial differential equations. In Biology, the chemotaxis is an important phenomenon, and the research on this topic has many applications. Chemotaxis-Fluid models have been used to model the chemotaxis in a viscous incompressible fluid. Taking into account the random environments, it is natural to consider the stochastic models. Since compared with the deterministic Chemotaxis-Fluid models, introducing stochastic noises will lead to new long-time behaviors, our project will contribute to the mathematical understanding of the chemotaxis and have potential applications.