Onset of Pattern Formation for the Stochastic Allen–Cahn Equation

We study the behavior of the solution of a stochastic Allen–Cahn equation $$\frac{\partial u_\varepsilon }{\partial t}=\frac{1}{2} \frac{\partial ^2 u_\varepsilon }{\partial x^2}+ u_\varepsilon -u_\varepsilon ^3+\sqrt{\varepsilon }\, \dot{W}$$ ∂ u ε ∂ t = 1 2 ∂ 2 u ε ∂ x 2 + u ε - u ε 3 + ε W ˙ , with Dirichlet boundary conditions on a suitably large space interval $$[-L_\varepsilon , L_\varepsilon ]$$ [ - L ε , L ε ] , starting from the identically zero function, and where $$\dot{W}$$ 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 ε 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 } Z ε , where $$\mathbb {Z}_\varepsilon =\mathbb {Z}$$ Z ε = Z modulo $$\lfloor \varepsilon ^{-1}L_\varepsilon \rfloor $$ ⌊ ε - 1 L ε ⌋ .