The Allen-Cahn equation with weakly critical random initial datum

• 成果类型: Article
• 署名作者: Gabriel, Simon; Rosati, Tommaso; Zygouras, Nikos
• 署名单位: University of Munster; University of Warwick
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01312-1
• 发表日期: 2025
• 页码: 1373-1446
• 关键词: kpz

摘要

This work considers the two-dimensional Allen-Cahn equation partial derivative tu=12 Delta u+mu-u3,u(0,x)=eta(x),for all(t,x)is an element of[0,infinity)xR2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t u = \frac{1}{2}\Delta u + \mathfrak {m}\, u -u<^>3, \quad u(0,x)= \eta (x), \qquad \forall (t,x) \in [0, \infty ) \times {\textbf {R}}<^>{2}, \end{aligned}$$\end{document}where the initial condition eta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \eta $$\end{document} is a two-dimensional white noise, which lies in the scaling critical space of initial data to the equation. In a weak coupling scaling, we establish a Gaussian limit with nontrivial size of fluctuations, thus casting the nonlinearity as marginally relevant. The result builds on a precise analysis of the Wild expansion of the solution and an understanding of the underlying stochastic and combinatorial structure. This gives rise to a representation for the limiting variance in terms of Butcher series associated to the solution of an ordinary differential equation.