Generation of fine transition layers and their dynamics for the stochastic Allen--Cahn equation

Abstract: We study an $\ep$-dependent stochastic Allen--Cahn equation with a mild random noise on a bounded domain in $\mathbb{R}^n$, $n\geq 2$. Here $\ep$ is a small positive parameter that represents formally the thickness of the solution interface, while the mild noise $\xi^\ep(t)$ is a smooth random function of $t$ of order $\mathcal O(\ep^{-\gamma})$ with $0<\gamma<1/3$ that converges to white noise as $\ep\rightarrow 0^+$. We consider initial data that are independent of $\ep$ satisfying some non-degeneracy conditions, and prove that steep transition layers---or interfaces---develop within a very short time of order $\ep^2|\ln\ep|$, which we call the "generation of interface". Next we study the motion of those transition layers and derive a stochastic motion law for the sharp interface limit as $\ep\rightarrow 0^+$. Furthermore, we prove that the thickness of the interface for $\ep$ small is indeed of order $\mathcal O(\ep)$ and that the solution profile near the interface remains close to that of a (squeezed) travelling wave, this means that the presence of the noise does not destroy the solution profile near the interface as long as the noise is spatially uniform. Our results on the motion of interface improve the earlier results of Funaki (1999) and Weber (2010) by considerably weakening the requirements for the initial data and establishing the robustness of the solution profile near the interface that has not been known before.