论文标题
噪音环境中的定向平均曲率流
Directed mean curvature flow in noisy environment
论文作者
论文摘要
我们考虑在弱高斯随机环境中平面上的定向平均曲率流。我们证明，从足够平坦的初始条件开始时，恢复的溶液会收敛到KPZ方程的Cole-HOPF溶液。该结果来自对使用规律性结构理论的不均匀噪声驱动的更通用的非线性SPDES系统的分析。然而，由于噪声的不均匀性，无法直接应用“黑匣子”结果[HAI14，BHZ19，CH16，BCCH21]，并且需要对无限二维规则性结构进行显着扩展。 对这种SPDES的一般系统的分析给出了两个更有趣的结果。首先，我们证明具有非常强的力的淬灭KPZ方程的解也会收敛到KPZ方程的Cole-HOPF溶液。其次，我们表明，在任何维度上，适当重新恢复和重新划定的淬火Edwards-Wilkinson模型会收敛到随机热方程。
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.
