In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application, we obtain the convergence of random attractors for non-autonomous stochastic reaction-diffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem. A differentiability condition on nonlinearity is omitted, which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity. These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.

中文翻译：

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弱收敛条件下非自治动力系统随机吸引子的上半连续性

在本文中，我们通过弱到弱限制代替通常的范数到范数限制来开发关于随机吸引子的上半连续性的标准。作为一个应用，当随机噪声的密度接近零时，我们获得了无界域上非自治随机反应扩散方程的随机吸引子的收敛性。利用Alaoglu弱紧性定理证明了解的弱收敛性。省略了非线性的可微性条件，这意味着随机吸引子的存在条件足以确保它们的上半连续性。这些结果极大地加强了文献中提出的上半连续性概念。