We develop a theory of mean-square random invariant manifolds for mean-square random dynamical systems generated by stochastic differential equations. This theory is applicable to stochastic partial differential equations driven by nonlinear noise. The existence of mean-square random invariant unstable manifolds is proved by the Lyapunov-Perron method based on a backward stochastic differential equation involving the conditional expectation with respect to a filtration. The existence of mean-square random stable invariant sets is also established but the existence of mean-square random stable invariant manifolds remains open.

中文翻译：

---

随机微分方程的均方随机不变流形

我们为随机微分方程生成的均方随机动力学系统开发了均方随机不变流形的理论。该理论适用于非线性噪声驱动的随机偏微分方程。通过Lyapunov-Perron方法，基于涉及过滤的条件期望的反向随机微分方程，证明了均方随机不变不稳定流形的存在。还建立了均方随机稳定不变集的存在，但是均方随机稳定不变流形的存在仍然是开放的。