The Allen–Cahn equation with weakly critical random initial datum

This work considers the two-dimensional Allen-Cahn equation ∂tu=12Δu+mu-u3,u(0,x)=η(x),∀(t,x)∈[0,∞)×R2, where the initial condition η 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.