Directed mean curvature flow in noisy environment

TL;DR

This paper proves convergence of a directed mean curvature flow in a noisy environment to the KPZ equation's Cole-Hopf solution, extending regularity structures to inhomogeneous noises and analyzing related SPDE systems.

Contribution

It extends the theory of regularity structures to handle inhomogeneous noises in infinite dimensions and proves convergence results for related stochastic PDEs.

Findings

Rescaled solutions converge to the Cole-Hopf solution of KPZ.

Quenched KPZ with strong force also converges to KPZ.

Rescaled quenched Edwards-Wilkinson model converges to stochastic heat equation.

Abstract

We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole-Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the "black box" result developed in the series of works [Hai14, BHZ19, CH16, BCCH21] cannot be applied directly and requires significant extension to infinite-dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole-Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards-Wilkinson model in any dimension converges to the stochastic heat equation.