噪声环境中的有向平均曲率流
Directed mean curvature flow in noisy environment
AI总结：
本文利用正则结构分析非均匀噪声驱动的非线性 SPDE，证明弱随机环境中有向平均曲率流在平坦初值下收敛到 KPZ 方程的 Cole-Hopf 解，并得到强外力 quenched KPZ 与任意维 quenched Edwards-Wilkinson 模型的极限结果。
摘要：
我们考虑平面上处于弱 Gaussian 随机环境中的有向平均曲率流。我们证明，当从足够平坦的初始条件出发时，经过重新缩放并重新中心化的解收敛于 KPZ 方程的 Cole-Hopf 解。这一结果源于对一个由非均匀噪声驱动的更一般非线性 SPDE 系统的分析，并使用了正则结构理论。然而，由于噪声的非均匀性，在系列工作 [Hai14, BHZ16, CH16, BCCH21] 中发展的“黑箱”结果不能直接应用，需要对无穷维正则结构进行重大扩展。对这一一般 SPDE 系统的分析还给出了另外两个有趣结果。首先，我们证明具有极强外力的 quenched KPZ 方程的解也收敛于 KPZ 方程的 Cole-Hopf 解。其次，我们表明，在任意维度中，经过适当重新缩放和重整化的 quenched Edwards-Wilkinson 模型收敛于随机热方程。
英文摘要：
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 [Hai14, BHZ19, CH16, BCCH21] 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.
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