We consider a family of random quasi-linear equations driven by nonlinear Wong–Zakai noise and parameterized by the non-zero size \(\lambda \) of noise. After proving the existence of a random attractor \(A_\lambda (\omega )\) in the square Lebesgue space, we then show that there is a residual dense subset of the space of nonzero real numbers such that, under the Hausdorff metric, the map \(\lambda \rightarrow A_\lambda (\theta _s\omega )\) is continuous at all points of the residual dense set, where \(\theta _s\) is a group of self-transformations on the probability space. We also prove that as \(\lambda \rightarrow \pm \infty \) the random attractor converges upper-semicontinuously to the global attractor of the deterministic quasi-linear equation. The upper semi-continuity result is new for nonlinear noise, while, the lower semi-continuity result is new even for linear noise. The theory of Baire category is the main tool used to prove the residual continuity.

中文翻译：

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具非线性噪声的拟线性方程组的随机吸引子的连续Wong-Zakai逼近

我们考虑一类由非线性Wong-Zakai噪声驱动并由非零大小\（\ lambda \）噪声参数化的随机准线性方程组。在证明方形Lebesgue空间中存在一个随机吸引子\（A_ \ lambda（\ omega）\）之后，我们证明了存在非零实数空间的剩余稠密子集，使得在Hausdorff度量下，映射\（\ lambda \ rightarrow A_ \ lambda（\ theta ss \ omega）\）在剩余密集集的所有点处都是连续的，其中\（\ theta ss）是概率空间上的一组自变换。我们也证明为\（\ lambda \ rightarrow \ pm \ infty \）随机吸引子上半连续收敛到确定性拟线性方程的整体吸引子。对于非线性噪声，较高的半连续性结果是新的，而对于线性噪声，较低的半连续性结果也是新的。Baire类理论是用来证明残差连续性的主要工具。