Onset of Pattern Formation for the Stochastic Allen–Cahn Equation

We study the behavior of the solution of a stochastic Allen–Cahn equation \(\frac{\partial u_\varepsilon }{\partial t}=\frac{1}{2} \frac{\partial ^2 u_\varepsilon }{\partial x^2}+ u_\varepsilon -u_\varepsilon ^3+\sqrt{\varepsilon }\, \dot{W}\) $\frac{\partial u_{\epsilon }}{\partial t}=\frac{1}{2}\frac{{\partial }^2{u}_{\epsilon }}{\partial x^2}+u_{\epsilon }-u_{\epsilon }^3+\sqrt{\epsilon }\dot{W}$ , with Dirichlet boundary conditions on a suitably large space interval \([-L_\varepsilon , L_\varepsilon ]\) $[-L_{\epsilon },L_{\epsilon }]$ , starting from the identically zero function, and where \(\dot{W}\) $\dot{W}$ 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 \(u_\varepsilon \) $u_{\epsilon }$ becomes close to 1 or \(-1\) $-1$ . 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 22:334–371, 1994) in the related context of a class of reaction–diffusion models obtained as a superposition of a speeded up stirring process and spin flip dynamics on \(\{-1,1\}^{\mathbb {Z}_\varepsilon }\) ${\{-1,1\}}^{{\mathbb{Z}}_{\epsilon }}$ , where \(\mathbb {Z}_\varepsilon =\mathbb {Z}\) ${\mathbb{Z}}_{\epsilon }=\mathbb{Z}$ modulo \(\lfloor \varepsilon ^{-1}L_\varepsilon \rfloor \) $\lfloor {\epsilon }^{-1}{L}_{\epsilon }\rfloor$ .