M. Gubinelli and N. Perkowski
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 Gonçalves 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 $t/12$.
M. Gubinelli and N. Perkowski. Energy solutions of KPZ are unique. Papers of Massimiliano Gubinelli, 2015, version 2. https://mgubi.org/papers/paper/kpz-energy-uniqueness/
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