On the space-time fluctuations of the SHE and KPZ equation in the entire $L^2$-regime for spatial dimensions $d\ge 3$

Abstract

We consider the mollified versions of the Kardar-Parisi-Zhang (KPZ) equation and the stochastic heat equation (SHE) in high dimensions $d\ge 3$ and analyze their probability distributions as the mollification is removed. Up to the $L^2$-criticality, we prove Gaussian limits, possibly with random perturbations, for the space-time fluctuations of the mollified versions around their stationarity. This result establishes a continuous analogue of the discrete case obtained by Cosco and Nakajima (2021), and further extends it to a multi-point framework.