Brownian fluctuations of the interface in a system with two linear attracted components and white noises

TL;DR

This paper analyzes the long-term behavior of interfaces in a two-component system governed by noisy Allen-Cahn equations with attractive interaction, showing that the interface center performs a Brownian motion over time.

Contribution

It demonstrates that, under small noise and initial conditions near an instanton, the interface center in a coupled Allen-Cahn system exhibits Brownian motion in the limit as noise vanishes.

Findings

Interface center follows Brownian motion over time.

Components remain close to a common instanton.

Results hold as noise parameter approaches zero.

Abstract

We concern the analysis of the long time behavior of interfaces in systems with two components. Each component evolves according to 1-d Allen-Cahn equation with Neumann boundary conditions, perturbed by small space-time white noise and with symmetric double well potential in the interval $[-\epsilon^{-1},\epsilon^{-1}]$. The two components interact with each other by an attractive linear force. Instantons are defined as the stationary solution of the Allen-Cahn equation without noise which connects two pure phases. We prove that for time $t=\epsilon^{-1}$, in the limit $\epsilon \rightarrow 0$, when initial states are close to an instanton, two components stay close to the same instanton, whose center moves as a Brownian motion.