在分布意义下，高斯白噪声可以看做是标准布朗运动的时间导数，因此Wong-Zakai逼近研究用简单过程逼近布朗运动时，其时间导数驱动的微分方程与对应的高斯白噪声驱动的微分方程的解之间的关系。随机动力系统理论就是从样本轨道的角度考虑解的演化趋势。应用随机动力系统理论于随机常微分方程的前提是随机常微分方程的解能够生成随机动力系统。主要技术是通过一个同胚映射将随机常微分方程转化为一个等价的随机微分方程。利用这种变换技巧的挑战在于这样的事实：所需的同胚映射的显式表达式不可缺少，但是对于实际中的系统这往往是很难甚至是无法做到的。对大多数随机偏微分方程来讲，这种共轭关系是否存在还是未知的，除了加性噪声和线性乘性噪声这种特殊情形。随机不变流形对于研究随机动力系统的长时期演变具有非常重要的意义。本项目分别研究加性噪声与线性乘性噪声驱动的非自治随机偏微分方程的解的Wong-Zakai逼近问题及稳定不变流形的。

In the sense of distribution, Guassian white noise is the time derivative of standard Brownian motion. Therefore, Wong-Zakai approximation investigate the relation between the corresponding solutions of differential equation driven by Gaussian white noise and differential equations driven by time derivative of simple processes, while the sequence of simple processes approximating Brownian motion. Random dynamical system(RDS) study the asymptotic evolution of the sample-wise solution of stochastic differential equations(SDEs). The prerequisite for applying the theory of RDS to study SDEs is to show that solutions to an underlying SDE generate an RDS. The main technique is to transform the SDE into an equivalent random differential equation by a homeomorphism. The challenges of using this transformation technique lie in the fact that an explicit expression for such a homeomorphism is needed, but difficult or impossible to construct explicitly for specific SDEs system from real world applications. In other words, although we can prove that such a homeomorphism exists, an explicit expression which allows us to work directly with the solutions of both equations may not be available. Random invariant manifold play an important role in studying the asymptotic evolution of RDS. This project focus on the Wong-Zakai approximation of solution and stable invariant manifold for nonautonomous stochastic partial differential equations with additive and linear multiplicative noise.