We study singular limits of stochastic evolution equations in the interplay of disappearing strength of the noise and insufficient regularity, where the equation in the limit with noise would not be defined due to lack of regularity. We recover previously known results on vanishing small noise with increasing roughness, but our main focus is to study for fixed noise the singular limit where the leading order differential operator in the equation may vanish. Although the noise is disappearing in the limit, additional deterministic terms appear due to renormalization effects. We separate the analysis of the equation from the convergence of stochastic terms and give a general framework for the main error estimates. This first reduces the result to bounds on a residual and in a second step to various bounds on the stochastic convolution. Moreover, as examples we apply our result to the singularly regularized Allen–Cahn (AC) equation with a vanishing Bilaplacian, and the Cahn–Hilliard/AC homotopy with space-time white noise in two spatial dimensions.

我们研究随机演化方程在噪声消失强度和规律性不足的相互作用中的奇异极限，其中由于缺乏规律性而无法定义噪声极限方程。我们恢复了先前已知的随着粗糙度增加而消失的小噪声的结果，但我们的主要重点是研究固定噪声的奇异极限，其中方程中的前阶微分算子可能会消失。尽管噪声在极限范围内消失，但由于重整化效应，出现了额外的确定性项。我们将方程的分析与随机项的收敛分开，并给出主要误差估计的一般框架。这首先将结果减少到残差的范围，然后在第二步中减少到随机卷积的各种范围。而且，