Onset of Pattern Formation for the Stochastic Allen–Cahn Equation

Stella Brassesco (), Glauco Valle () and Maria Eulália Vares ()
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Stella Brassesco: Instituto Venezolano de Investigaciones Científicas
Glauco Valle: Universidade Federal do Rio de Janeiro
Maria Eulália Vares: Universidade Federal do Rio de Janeiro

Journal of Theoretical Probability, 2025, vol. 38, issue 1, 1-23

Abstract: Abstract 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 ⌋ .

Keywords: Stochastic Allen–Cahn equation; Reaction–diffusion models; Phase separation; 60H15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-024-01395-5

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