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Interface fluctuations for $1$D stochastic Allen-Cahn equation revisited

Weijun Xu, Wenhao Zhao, Shuhan Zhou

TL;DR

The paper analyzes interface fluctuations for the one-dimensional stochastic Allen–Cahn equation with small space-time white noise, showing that starting near the standing wave manifold, the solution remains close to translated standing waves and the interface location converges to an explicit Itô diffusion on the long time scale $\varepsilon^{-2\gamma-1}$. The authors develop a novel analytic framework built on a hierarchy of functional correctors that cancels potential divergences and on a systematic decomposition of Fréchet derivatives of the deterministic flow, enabling the full small-noise regime to be treated. They derive an explicit limiting SDE for the interface position with coefficients $\alpha_1$ and $\alpha_2$, and establish sharp control of the deterministic flow and its higher-order derivatives through precise kernel decompositions. The work also provides a robust methodological framework potentially applicable to other singular SPDEs, offering both a sharp interface limit and tools for long-time analysis of stochastic perturbations of reaction-diffusion dynamics.

Abstract

We revisit the interface fluctuation problem for the $1$D 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 \cite{Fun95} to full small noise regime, and recovers the result in \cite{BBDMP98}. The proof builds on the analytic framework in \cite{Fun95}. Our main novelty is the construction of a series of functional correctors that are designed to recursively cancel potential divergences. Moreover, to show 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.

Interface fluctuations for $1$D stochastic Allen-Cahn equation revisited

TL;DR

The paper analyzes interface fluctuations for the one-dimensional stochastic Allen–Cahn equation with small space-time white noise, showing that starting near the standing wave manifold, the solution remains close to translated standing waves and the interface location converges to an explicit Itô diffusion on the long time scale . The authors develop a novel analytic framework built on a hierarchy of functional correctors that cancels potential divergences and on a systematic decomposition of Fréchet derivatives of the deterministic flow, enabling the full small-noise regime to be treated. They derive an explicit limiting SDE for the interface position with coefficients and , and establish sharp control of the deterministic flow and its higher-order derivatives through precise kernel decompositions. The work also provides a robust methodological framework potentially applicable to other singular SPDEs, offering both a sharp interface limit and tools for long-time analysis of stochastic perturbations of reaction-diffusion dynamics.

Abstract

We revisit the interface fluctuation problem for the D 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 \cite{Fun95} to full small noise regime, and recovers the result in \cite{BBDMP98}. The proof builds on the analytic framework in \cite{Fun95}. Our main novelty is the construction of a series of functional correctors that are designed to recursively cancel potential divergences. Moreover, to show 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.
Paper Structure (21 sections, 43 theorems, 404 equations)

This paper contains 21 sections, 43 theorems, 404 equations.

Table of Contents

  1. Introduction
  2. Convergence
  3. Overview of the proof
  4. Construction of the functional correctors
  5. Convergence of the interface process -- proof of Theorem \ref{['thm:main']}
  6. Analysis of the deterministic flow
  7. Strategy of the proof
  8. Preliminaries
  9. Linearized PDE
  10. Kernel decompositions
  11. Fréchet derivatives of the deterministic flow
  12. Proof of Theorems \ref{['thm:derivativesOfPsicork']}, \ref{['thm:zeta_continuity_bound']} and \ref{['thm:functional_equation']}
  13. Fréchet derivatives of the limiting functional $\zeta$
  14. Behavior of the correctors
  15. Proof of Theorem \ref{['thm:derivativesOfPsicork']}
  16. ...and 6 more sections

Key Result

Theorem 1.2

Fix $\gamma>0$ and $\kappa \in (0,\gamma)$ arbitrary. Let $\xi_0\in \mathbb{R}$, and $u_\varepsilon$ be the solution to e:main_eqn with initial data $u_\varepsilon[0] = m_{\xi_0 / \sqrt{\varepsilon}} \in \mathcal{M}$. Let $v_{\varepsilon}(t,x) := u_\varepsilon(\varepsilon^{-2\gamma-1}t,x)$. Then the

Theorems & Definitions (88)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • ...and 78 more