We study the random motion of interfaces in stochastic Allen–Cahn equations. In one dimension, we show that, on a suitable long time scale, the solution stays close to the family of standing wave solutions, while the interface location converges to a diffusion. Our main novelty is the construction of functional correctors to cancel potential divergences. We also establish an analogous result for more singular noise leading to distribution-valued solutions. In this setting, we show that two divergent quantities arising from renormalization and Itô’s formula cancel each other.
In higher dimensions, we construct stochastic mean curvature flow driven by noise that is white in time and smooth in space, and identify it as the sharp interface limit of the stochastic Allen–Cahn equation with the same noise slightly smoothed in time. To construct this flow, we use a transformation developed by Buckdahn, Keller, Ma and Zhang (2020), which reduces the stochastic equation to a classical PDE with random coefficients.
This talk is based on joint works with Weijun Xu and Wenhao Zhao.