the physics and mathematics community. In particular, the *weak KPZ universality conjecture* [@Corwin12; @QuaS; @Sp16] states that the fluctuations of a large class of one-dimensional microscopic interface growth models are ruled at the macroscopic scale by solutions of the *KPZ equation*, which reads as follows: consider a time variable $t$ and a one-dimensional space variable $u$, then the evolution of the height ${{\mathcal Z}}(t,u)$ of the randomly growing interface can be described by $$d{{\mathcal Z}}(t,u) = A \Delta {{\mathcal Z}}(t,u)\; dt + B \big( \nabla {{\mathcal Z}}(t,u)\big)^2 \; dt + \sqrt C d {{\mathcal W}}_t, \label{eq:KPZintro}$$ where $A,B,C$ are thermodynamic constants depending on the model and $\partial_t{{\mathcal W}}$ is a space-time white noise. Note that the non-linearity $( \nabla {{\mathcal Z}}(t,u))^2$ makes the KPZ equation ill-posed, essentially because the trajectory of the solution lacks space regularity (due to the presence of the white noise), and therefore the square of its distributional derivative is not defined.

One possible way to solve this equation is to consider its *Cole-Hopf transformation* $\Phi$, which solves a stochastic heat equation with a multiplicative noise (SHE) and is related to ${{\mathcal Z}}$ through a $\log$–transformation. Since the SHE is linear, its solution can easily be constructed 