本项目主要研究随机Swift-Hohenberg方程（SHE）的遍历性。SHE源于热对流现象，是非线性抛物型偏微分方程，常用来刻画模式形成行为。考虑到随机因素的影响，在方程中增加噪声项，就构成了随机SHE。遍历性是随机偏微分方程解的一种长时间行为。我们首先证明随机SHE解的存在唯一性，之后在解的马氏性基础上定义半群，研究马氏半群的不变测度的存在和唯一性。进一步得到解的遍历性以及相应的Kolmogorov方程解的性质。由于有界和无界区域时，状态空间的巨大差异性，对这两种情形分别进行研究。以随机分析和泛函分析为主要工具，借鉴和参考随机Navier-Stokes方程以及确定性SHE的方法和结论，最终揭示随机SHE解的稳定性。通过本项目的研究，既可以明确随机和确定性SHE的区别和联系，又能指引对其他类型的随机偏微分方程的研究；最后，对比有界与无界区域时的方法和结论，更加深刻理解其中的差异和根源。

We are concerned to the ergodicity of stochastic Swift-Hohenberg equations (for short, SHE). SHE was derived from the equations for thermal convection. It is a non-linear parabolic partial differential equations, which is used to characterize the pattern-forming behavior. Due to the impacts of random factors, we add a noise term in our SHE, and then we get a stochastic SHE. Ergodicity is an important long-time behavior of stochastic partial differential equations. At first, we need to show the existence and uniqueness of the solution of stochastic SHE. Later, we can define a semi-group based on the Markov property of our solutions. After these preparations, we prove that our Markov semi-group has a unique invariant measure. What’s more, we get the ergodicity of our solutions and the properties of the corresponding Kolmogorov equations. Using stochastic calculus and functional analysis, with the help of the results and methods from stochastic Navier-Stokes equations and deterministic SHE, we aim to expose the stability of stochastic SHE. Because the big difference between bounded domains and unbounded ones, we study these two cases separately. On the one hand, this program can help us to investigate the other kinds of stochastic partial differential equations; on the other hand, we can learn more details about the difference between stochastic and deterministic cases of SHE. In the end, comparing the methods and results of bounded and unbounded cases, we can understand the big difference between situations more clearly.