Mesoscopic averaging of the two-dimensional KPZ equation

Abstract

We study the limit of a local average of the KPZ equation in dimension $d=2$ with general initial data in the subcritical regime. Our result shows that a proper spatial averaging of the KPZ equation converges in distribution to the sum of the solution to a deterministic KPZ equation and a Gaussian random variable that depends solely on the scale of averaging. This shows a unique mesoscopic averaging phenomenon that is only present in dimension two. Our work is inspired by the recent findings by Chatterjee \cite{chatterjee2021weak}.

arXiv comment27 pages, accepted version, fixed typos, added a few references, and improved presentation

References in corpus (8)

• The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher
• On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window
• Moments of the 2D SHE at criticality
• Renormalizing the Kardar-Parisi-Zhang equation in $d\ge 3$ in weak disorder
• A forward-backward SDE from the 2D nonlinear stochastic heat equation
• Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two
• The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos
• Convergence of deterministic growth models