Scaling limit of the KPZ equation with non-integrable spatial correlations

TL;DR

This paper investigates the large-scale behavior of the KPZ equation with non-integrable spatial correlations in dimensions d ≥ 3, showing it converges to a correlated stochastic heat equation with preserved spatial correlations.

Contribution

It demonstrates that non-integrable spatial correlations in the KPZ equation lead to a limiting additive stochastic heat equation with persistent spatial correlations, contrasting with the compact support case.

Findings

Scaling limit is described by a correlated stochastic heat equation.

The noise in the limit retains the spatial correlation structure.

Convergence occurs in probability under a suitable coupling.

Abstract

We study the large scale fluctuations of the KPZ equation in dimensions $d \geq 3$ driven by Gaussian noise that is white in time Gaussian but features non-integrable spatial correlation with decay rate $\kappa \in (2, d)$ and a suitable limiting profile. We show that its scaling limit is described by the corresponding additive stochastic heat equation. In contrast to the case of compactly supported covariance, the noise in the stochastic heat equation retains spatial correlation with covariance $|x|^{-\kappa}$. Surprisingly, the noise driving the limiting equation turns out to be the scaling limit of the noise driving the KPZ equation so that, under a suitable coupling, one has convergence in probability, unlike in the case of integrable correlations where fluctuations are enhanced in the limit and convergence is necessarily weak.