Generalised stochastic curvature flow in $d \geq 2$, and sharp interface limit for the stochastic Allen-Cahn equation with nonlinear diffusion

Abstract

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.