论文标题
随机方程的单数极限
Singular limits for stochastic equations
论文作者
论文摘要
我们研究了随机演化方程在消失的噪声和规律性不足的相互作用中的单数极限，在这种情况下，由于缺乏规律性，无法定义噪声的方程式。我们恢复了以前恢复以增加粗糙度消失的噪声的先前已知结果，但我们的主要重点是研究固定噪声的奇异极限，而方程式中领先的差异操作员可能会消失。尽管噪声在极限内消失，但由于重新归一化的效果而出现了其他确定性项。我们将方程式的分析与随机项的收敛性分析，并为主要误差估计提供了一般框架。这首先将结果降低到残留的边界，然后在第二步中降低到随机卷积的各个边界。此外，作为示例，我们将结果应用于具有消失的双乳杆菌的单个正规化的Allen-Cahn方程，以及在两个空间维度中带有时空白噪声的Cahn-Hilliard/Allen-Cahn同型。
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 a singularly regularized Allen-Cahn equation with a vanishing Bilaplacian, and the Cahn-Hilliard/Allen-Cahn homotopy with space-time white noise in two spatial dimensions.
