and it is unique, and the solution to the KPZ equation can then simply be defined as the inverse Cole-Hopf transformation of $\Phi$. However, the solution to the SHE is too irregular to allow for a change of variable formula, and a priori there is no meaningful equation associated to its inverse Cole-Hopf transformation. Only recently Hairer has developed a meaningful notion of solution for the KPZ equation and proved existence and uniqueness of such solutions with periodic boundary conditions, see [@Hairer2013]. His approach uses rough path integrals to construct the nonlinearity $(\nabla {{\mathcal Z}}(t,u))^2$, and it has inspired the development of new technologies (regularity structures [@Hairer2014] and paracontrolled distributions [@Gubinelli2015Paracontrolled]) for the so-called *singular* stochastic partial differential equations (SPDEs).

The first breakthrough towards the weak KPZ universality conjecture is due to Bertini and Giacomin: in their seminal paper [@BerG], they show that the Cole-Hopf solution can be obtained as a scaling limit of the *weakly asymmetric exclusion process* (WASEP) (which will be defined ahead). Their approach consists in performing the Cole-Hopf transformation at the microscopic level, following [@GAR], and then showing that this microscopic Cole-Hopf transformation solves a linear equation (similarly to what happens at the macroscopic level). Since 