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

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 ≥ 3. In this case, the non-linearity formally vanishes on large scales under the diffusive rescaling. 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 in [] in the critical case. However, the result in [] is not a fluctuation result but rather an L 2 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.