Existence, scaling, and spectral gap for traveling fronts in the 2D renormalized Allen--Cahn equation

2025/12/16 by Gideon Chiusole, Christian Kuehn, Chiusole, Gideon +1
Computer Science · Economics, Econometrics and Finance · Materials Science · #34L15 #35C07 #37L15 #47A53 #47A56 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Dynamics and Pattern Formation #Solidification and crystal growth phenomena #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2512.14245

openalex publication_date 2025/12/16 · openalex created_date 2025/12/18 · openalex updated_date 2026/07/28

Abstract

We study the deterministic skeleton of the renormalized stochastic Allen--Cahn equation in spatial dimension 2. For all sufficiently small regularization parameters δ>0, we construct monotone traveling wave front solutions connecting the renormalized equilibria, derive a small-δ asymptotic description of their profile and speed, and identify the leading-order contributions. Linearizing about the wave and working in a naturally chosen weighted space, we show that there exists a spectral gap between the symmetry induced eigenvalue 0 and the rest of the spectrum. The spectral gap grows linearly in the renormalization constant as δ\downarrow 0.

Citations

• Local Phase Tracking and Metastability of Planar Waves in Stochastic Reaction-Diffusion Systems
• Travelling Waves in Monostable and Bistable Stochastic Partial\n Differential Equations

Related

• Exponential loss of memory for the 2-dimensional Allen-Cahn equation\n with small noise
• Sharp convergence to steady states of Allen-Cahn
• Interface fluctuations for 1D stochastic Allen-Cahn equation -- singular regime
• Interface fluctuations for 1D stochastic Allen-Cahn equation revisited
• The Allen-Cahn equation with weakly critical random initial datum