Central limit theorem for the Allen-Cahn equation with supercritical random initial conditions

TL;DR

This paper proves a central limit theorem for the Allen-Cahn equation with supercritical Gaussian initial data, showing convergence to a heat equation solution driven by white noise in high dimensions.

Contribution

It establishes a new fluctuation result for supercritical initial conditions, extending previous work and introducing novel techniques involving comparison principles and Malliavin calculus.

Findings

Rescaled solutions converge to a heat equation with white noise initial conditions.

The limit depends on the randomness source and non-linearity.

New proof methods are applicable to other supercritical stochastic PDEs.

Abstract

We study the large-scale behavior of solutions to the Allen-Cahn reaction-diffusion equation with Gaussian initial data. We consider the case of short-range dependence in the associated supercritical regime with spatial dimension $d \ge 3$. Under diffusive rescaling, the non-linearity formally vanishes on large scales in this case. Accordingly, we prove a central limit theorem for the rescaled solution, more precisely, that it converges to the solution of the heat equation started from a white noise. These initial conditions for the limit depend non-trivially both on the source of randomness and on the non-linearity. Our proof uses estimates obtained by a combination of comparison principles and Malliavin calculus, initiated by Castillo and Dunlap in arXiv:2509.06260 in the critical case. However, the result there is not a fluctuation result but rather an $L^2_\mathbf{P}$ comparison to a McKean-Vlasov problem with Gaussian solutions. Hence the mechanism behind the Gaussianity of the limit differs, and the proof requires new ideas that should be further applicable to other supercritical problems.