Onset of pattern formation for the stochastic Allen-Cahn equation

Abstract

We study the behavior of the solution of a stochastic Allen-Cahn equation $\frac{\partial u_\eps }{\partial t}=\frac 12 \frac{\partial^2 u_\eps }{\partial x^2}+ u_\eps -u_\eps^3+\sqrt\eps\, \dot W$, with Dirichlet boundary conditions on a suitably large space interval $[-L_\eps , L_\eps]$, starting from the identically zero function, and where $\dot W$ is a space-time white noise. Our main goal is the description, in the small noise limit, of the onset of the phase separation, with the emergence of spatial regions where $u_\eps$ becomes close $1$ or $-1$. The time scale and the spatial structure are determined by a suitable Gaussian process that appears as the solution of the corresponding linearized equation. This issue has been initially examined by De Masi et al. [Ann. Probab. {\bf 22}, (1994), 334-371] in the related context of a class of reaction-diffusion models obtained as a superposition of a speeded up stirring process and a spin flip dynamics on $\{-1,1\}^{\mathbb{Z}_\eps}$, where $\mathbb{Z}_\eps=\mathbb{Z}$ modulo $\lfloor\eps^{-1}L_\eps\rfloor$.