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<p>We revisit the interface fluctuation problem for the 1D Allen-Cahn equation perturbed by a small space-time white noise. We show that if the initial data is a standing wave solution to the deterministic equation, then under proper long time scale, the solution is still close to the family of traveling wave solutions. Furthermore, the motion of the interface converges to an explicit stochastic differential equation. This extends the classical result in Funaki (Probab Theory Relat Fields 102(2):221–288, 1995) to a full small noise regime, and recovers the result in Brassesco et al. (J Theor Probab 11:25–80, 1998). The proof builds on the analytic framework in Funaki (Probab Theory Relat Fields 102(2):221–288, 1995). Our main novelty is the construction of a series of functional correctors that are designed to recursively cancel potential divergences. Moreover, to show that these correctors are well-behaved, we develop a systematic decomposition of Fréchet derivatives of the deterministic Allen-Cahn flow of all orders. This decomposition is of its own interest, and may be useful in other situations as well.</p>

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Interface Fluctuations for 1D Stochastic Allen-Cahn Equation Revisited

  • 期刊论文

  • 发表日期：

    2025年8月18日

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  • Weijun Xu,
  • Wenhao Zhao,
  • Shuhan Zhou

摘要

We revisit the interface fluctuation problem for the 1D Allen-Cahn equation perturbed by a small space-time white noise. We show that if the initial data is a standing wave solution to the deterministic equation, then under proper long time scale, the solution is still close to the family of traveling wave solutions. Furthermore, the motion of the interface converges to an explicit stochastic differential equation. This extends the classical result in Funaki (Probab Theory Relat Fields 102(2):221–288, 1995) to a full small noise regime, and recovers the result in Brassesco et al. (J Theor Probab 11:25–80, 1998). The proof builds on the analytic framework in Funaki (Probab Theory Relat Fields 102(2):221–288, 1995). Our main novelty is the construction of a series of functional correctors that are designed to recursively cancel potential divergences. Moreover, to show that these correctors are well-behaved, we develop a systematic decomposition of Fréchet derivatives of the deterministic Allen-Cahn flow of all orders. This decomposition is of its own interest, and may be useful in other situations as well.