A pullback random attractor for a cocycle is a family of compact invariant attracting random sets $A(t, \theta_s \cdot)$, where $(t, s)$ is a point of the Euclid plane and $\theta$ is a group of measure-preserving transformations on a probability space. Under three conditions including the union closedness of the universe, the time-sample compactness of the PRA and the joint continuity of the cocycle, we prove that the map $(t, s) \to A(t, \theta_s \cdot)$ is continuous at all points of a residual diagonal-closed subset of the Euclid plane and full pre-continuous with respect to the Hausdorff metric. Applying to the non-autonomous stochastic $g$‑Navier–Stokes equation, we show the sample-continuity and local-uniform asymptotic compactness of the cocycle, which lead to the existence, residual continuity and pre-continuity of a PRA.

中文翻译：

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随机 $g$-Navier-Stokes 方程的回调随机吸引子的几乎连续性

Cocycle 的回拉随机吸引子是一组紧致不变吸引随机集 $A(t, \theta_s \cdot)$，其中 $(t, s)$ 是欧几里得平面的一个点，$\theta$ 是一个概率空间上的一组保持测量的变换。在宇宙的联合封闭性、PRA 的时间样本紧性和环的联合连续性三个条件下，我们证明了映射 $(t, s) \to A(t, \theta_s \cdot)$在欧几里德平面的残差对角闭合子集的所有点上都是连续的，并且相对于豪斯多夫度量是完全预连续的。应用于非自治随机$g$-Navier-Stokes方程，我们展示了cocycle的样本连续性和局部均匀渐近紧致性，这导致了PRA的存在、残差连续性和预连续性。