Interface fluctuations for $1$D stochastic Allen-Cahn equation -- singular regime

Abstract

We study interface fluctuations for the $1$D stochastic Allen-Cahn equation perturbed by half a spatial derivative of the spacetime white noise. This half derivative makes the solution distribution-valued, so that proper renormalization is needed to make sense of the solution. We show that if the noise is sufficiently small, then an analogue of the classical results by \cite{Fun95,BBDMP98} holds in this singular regime. More precisely, for initial data close to the traveling wave solution of the deterministic equation, under proper long time scaling, the solution still stays close to the family of traveling waves, and the interface location moves according to an approximate diffusion process. There is one interesting difference between our singular regime and the classical situation: even if the solution and its approximate phase separation point are both well defined, the intended diffusion describing the movement of the canonical candidate of the phase point is not (even for fixed $\eps$). Two infinite quantities arise from the derivation of such an SDE, one due to singularity of the noise, and the other from renormalization. Magically, it turns out that they cancel out each other, thus making the derivation of the interface SDE valid in the $\eps \rightarrow 0$ limit.