In this paper we discuss the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for stochastic Ginzburg-Landau equations driven by a white noise. We first apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions. Consequently, we show that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. At last, when the stochastic Ginzburg-Landau equation is driven by a linear multiplicative noise, we establish the convergence of solutions of Wong-Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.

中文翻译：

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随机Ginzburg-Landau方程的Wong-Zakai逼近和渐近行为

在本文中，我们讨论了由平稳过程通过Wiener位移给出的Wong-Zakai逼近及其与白噪声驱动的随机Ginzburg-Landau方程相关的长期路径行为。我们首先应用Galerkin方法和紧密性参数来证明弱解的存在性和唯一性。因此，我们证明了近似方程在比原始随机方程弱得多的条件下具有回撤随机吸引子。最后，在线性乘性噪声驱动下随机Ginzburg-Landau方程时，随着近似大小接近零，我们建立了Wong-Zakai近似解的收敛性和近似随机系统的随机吸引子的上半连续性。