This paper studies the long time stability of both stochastic heat equations on a bounded domain driven by a correlated noise and their approximations. It is popular for researchers to prove the intermittency of the solution which means that the moments of solution to stochastic heat equation usually grow exponentially to infinite and this hints that the solution to stochastic heat equation is generally not stable in long time. However, quite surprisingly in this paper we show that when the domain is bounded and when the noise is not singular in spatial variables, the system can be long time stable and we also prove that we can approximate the solution by its finite dimensional spectral approximation which is also long time stable. The idea is to use eigenfunction expansion of the Laplacian on bounded domain. We also present numerical experiments which are consistent with our theoretical results.

翻译：本文研究有界域内由相关噪声驱动的随机热方程及其逼近格式的长时间稳定性。学界普遍通过证明解的间歇性来揭示随机热方程矩通常呈指数增长至无穷，这意味着随机热方程在长时间尺度上通常不稳定。然而令人惊讶的是，本文表明当区域有界且噪声在空间变量上非奇异时，系统可实现长时间稳定，同时证明其有限维谱逼近解亦具有长时间稳定性。核心思想是利用有界域拉普拉斯算子的特征函数展开。文中还给出了与理论结果一致的数值实验验证。