This paper is mainly concerned with the asymptotic dynamics of non-autonomous stochastic 3D globally modified Navier–Stokes equations driven by nonlinear noise. Based on the well-posedness of such equations, we first show the existence and uniqueness of weak pullback mean random attractors. Then we investigate the existence of (periodic) invariant measures, the zero-noise limit of periodic invariant measures and their limit as the modification parameter \(N\rightarrow N_0\in (0,+\infty )\). Furthermore, under weaker conditions, we obtain the existence of invariant measures as well as their limiting behaviors when the external term is independent of time. Finally, by using weak convergence method, we establish the large deviation principle for the solution processes.

本文主要研究非线性噪声驱动的非自治随机3D全局修正纳维-斯托克斯方程的渐近动力学。基于此类方程的适定性，我们首先证明了弱回调均值随机吸引子的存在性和唯一性。然后我们研究（周期性）不变测度的存在性、周期性不变测度的零噪声极限及其作为修改参数的极限\(N\rightarrow N_0\in (0,+\infty )\)。此外，在较弱的条件下，当外部项与时间无关时，我们获得了不变测度的存在及其限制行为。最后，利用弱收敛方法，建立了求解过程的大偏差原理。