Abstract

We study the Kardar-Parisi-Zhang equation in dimension $d\geq 3$ with space-time white noise which is smoothed in space. There is a natural disorder parameter attached to this equation which measures the intensity of the noise. We show that when the disorder is small, the approximating solution converges to a well-defined limit (with the limit depending on both the disorder and the mollification procedure), while the re-scaled fluctuations converge to a Gaussian limit as predicted by the Edwards-Wilkionson regime.

We also study the associated stochastic heat equation with multiplicative noise, which carries a natural Gaussian mutiplicative noise (GMC) on the Wiener space. When the disorder is large, we also show that the total mass of the GMC converges to zero, while the endpoint distribution of a Brownian path under the (renormlaized) GMC measure is purely atomic.

Based on joint works with, A. Shamov & O. Zeitouni, F. Comets & C. Cosco as well Y. Broeker.