# Renormalizing the Kardar-Parisi-Zhang equation in $d\geq 3$ in weak disorder

## TL;DR

This paper investigates the behavior of the KPZ equation in three or more dimensions under weak noise, identifying its limiting distribution and establishing tail bounds and moment existence.

## Contribution

It characterizes the limiting distribution of the KPZ equation in higher dimensions with weak noise and provides strong approximations and tail bounds.

## Key findings

- Limit law identified for the KPZ equation in $d\,\geq\,3$
- Limit has sub-Gaussian lower tails
- All moments exist for the limit distribution

## Abstract

We study Kardar-Parisi-Zhang equation in spatial dimension 3 or larger driven by a Gaussian space-time white noise with a small convolution in space. When the noise intensity is small, it is known that the solutions converge to a random limit as the smoothing parameter is turned off. We identify this limit, in the case of general initial conditions ranging from flat to droplet. We provide strong approximations of the solution which obey exactly the limit law. We prove that this limit has sub-Gaussian lower tails, implying existence of all negative (and positive) moments.