We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite-dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.

中文翻译：

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噪声环境下的定向平均曲率流

我们考虑弱高斯随机环境中平面上的有向平均曲率流。我们证明，当从足够平坦的初始条件开始时，重新调整和重新调整的解收敛到 KPZ 方程的 Cole-Hopf 解。这一结果是使用正则结构理论对由非均匀噪声驱动的更一般的非线性 SPDE 系统进行分析得出的。然而，由于噪声的不均匀性，该系列工作中开发的“黑匣子”结果不能直接应用，需要对无限维正则结构进行重大扩展。对这个 SPDE 通用系统的分析给出了两个更有趣的结果。首先，我们证明具有很强力的淬火 KPZ 方程的解也收敛于 KPZ 方程的 Cole-Hopf 解。其次，我们证明了任何维度上经过适当重新调整和重新归一化的淬火爱德华兹-威尔金森模型都收敛于随机热方程。