then, this strategy has been used in more sophisticated models, see [@Corwin2016; @CST; @corwin2017; @Dembo2016], however the applicability of the microscopic Cole-Hopf transformation is limited to a very specific class of particle systems.

Another way to look at the KPZ equation is via the *stochastic Burgers equation* (SBE), which is obtained from by taking its derivative: if ${{\mathcal Y}}_t = \nabla {{\mathcal Z}}_t$, then ${{\mathcal Y}}_t$ satisfies $$\label{eq:SBEintro}
d{{\mathcal Y}}(t,u) = A \Delta {{\mathcal Y}}(t,u)\; dt + B\nabla \big( {{\mathcal Y}}^2(t,u)\big) \; dt + \sqrt C d\nabla{{\mathcal W}}_t,$$ which has of course the same regularity issues as the KPZ equation. Nevertheless, this formulation is well adapted to derive KPZ behavior from microscopic models. Indeed, the work initiated by Gonçalves and Jara in [@gj2014] has introduced a new tool, called *second order Boltzmann-Gibbs principle* (BGP), which makes the non-linear term $\nabla ( {{\mathcal Y}}^2(t,u))$ of the SBE naturally emerge from the underlying microscopic dynamics. The authors have first proved the BGP for general weakly asymmetric simple exclusion processes, and shortly thereafter it has been extended to a wider class of microscopic systems, such as zero-range models [@GJS], integrable Hamiltonian one-dimensional chains [@GJSi], non-degenerate kinetically constrained models [@blondel2016], exclusion processes with defects 