Energy solutions of KPZ are unique

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 $t/12$.