Energy solutions of KPZ are unique

TL;DR

This paper proves the uniqueness of energy solutions for the KPZ equation on the real line, confirming the weak universality conjecture and revealing a fundamental difference from the Cole-Hopf solution.

Contribution

It provides the first intrinsic well-posedness proof for KPZ energy solutions, establishing their uniqueness and implications for universality.

Findings

Energy solutions are unique for KPZ on the real line.

The energy solution differs from the Cole-Hopf solution by an additional drift.

Supports the weak KPZ universality conjecture.

Abstract

The Kardar-Parisi-Zhang (KPZ) equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth. Here we provide the first intrinsic well-posedness result for the KPZ equation on the real line by showing that its energy solutions (as introduced by Gon\c{c}alves and Jara and later refined by Gubinelli and Jara) are unique. Together with various convergence results already present in the literature, this establishes the weak KPZ universality conjecture for a wide class of models. A remarkable consequence is that the energy solution to the KPZ equation is not equal to the Cole-Hopf solution, but it involves an additional drift $t/12$.