dynamics changes the boundary conditions to Robin’s type, $\nabla \Phi_t(0) = -D \Phi_t(0)$, $\nabla \Phi_t(1) = D \Phi_t(1)$ for a constant $D \in {{\mathbb R}}$ which depends on the parameters $A,B,C$.

In that sense ${{\mathcal Z}}$ can be interpreted as a Cole-Hopf solution to the KPZ equation with inhomogeneous Neumann boundary conditions, and then the question arises which of the two formulations (inhomogeneous or homogeneous Neumann conditions) accurately describes the behavior of our stochastic process at the boundary. While of course the main difficulty with the KPZ equation is that its solutions are non-differentiable and in particular we cannot evaluate $\nabla {{\mathcal Z}}_t(0)$ pointwise, we show in Proposition \[prop:boundary\] that after averaging a bit in time there are canonical interpretations for $\nabla {{\mathcal Z}}_t(0)$ and $\nabla {{\mathcal Z}}_t(1)$, and both indeed vanish. We also show in Proposition \[prop:CH-boundary\] that the formal change of the boundary conditions from Neumann to Robin in the exponential transformation $\Phi = e^{{{\mathcal Z}}}$ is reflected in the “pointwise” boundary behavior of $\Phi$ (again after averaging in time).

This should be compared with the recent work of Gerencsér and Hairer [@Gerencser2017] who show that the classical solution ${{\mathcal Z}}^\varepsilon$ to the KPZ equation, $$\label{eq:GH-KPZ-intro}
   d{{\mathcal Z}}^\varepsilon_t 