d≥2 下的广义随机曲率流，以及带非线性扩散的随机 Allen-Cahn 方程的尖锐界面极限
Generalised stochastic curvature flow in $d \geq 2$, and sharp interface limit for the stochastic Allen-Cahn equation with nonlinear diffusion
中文总结 AI 辅助
本文构造了d≥2下受时间白噪声驱动的广义随机曲率流局部时间解，将其应用于带非线性扩散的随机Allen-Cahn方程，证明该方程的尖锐界面极限为对应随机曲率流。
中文摘要 AI 辅助
我们构造了维数 d≥2 下由时间白噪声、空间光滑高斯噪声驱动的广义各向异性依赖方向的曲率流的局部时间解。这似乎是首个同时允许空间依赖的时间白噪声解构造，即使在更简单的各向异性随机平均曲率流情形下亦是如此。主要难点在于描述该流的随机偏微分方程具有乘性噪声，其非线性依赖于解及其梯度。关键技术是基于粗糙特征的变换（由文献[BKMZ20]提出），该变换可消除此粗糙乘性项。我们还阐明了该变换与先前已知特殊情形的关系。作为该构造的应用，我们证明在短时间区间内，带非线性扩散且具有相同噪声（时间上略作光滑处理）的随机 Allen-Cahn 方程的尖锐界面极限，由上述依赖方向的随机曲率流给出。
英文摘要
We construct the local-in-time solution of the generalised anisotropic direction-dependent curvature flow in dimension $d \geq 2$ forced by a white-in-time and smooth-in-space Gaussian noise. This seems to be the first construction with a white-in-time noise which also allows spatial dependence, even in the simpler case of isotropic stochastic mean curvature flow. The main difficulty is that the stochastic PDE describing the flow has a multiplicative noise depending nonlinearly on both the solution and its gradient. The key technique is a transform developed in \cite{BKMZ20} based on rough characteristics that removes this rough multiplicative term. We also illustrate the relationship of this transform with previously known special situations. As an application of the construction, we show that in a short time interval, the sharp interface limit of the stochastic Allen-Cahn equation with nonlinear diffusion and the same noise (slightly smoothened in time) is given by the above direction-dependent stochastic curvature flow.
