= \Delta {{\mathcal Z}}^\varepsilon_t dt + \big((\nabla{{\mathcal Z}}^\varepsilon_t)^2 - c^\varepsilon\big) dt +  d {{\mathcal W}}^\varepsilon_t \;,$$ with Neumann boundary condition $\nabla {{\mathcal Z}}^\varepsilon_t(0) = \nabla {{\mathcal Z}}^\varepsilon_t(1) = 0$ and where ${{\mathcal W}}^\varepsilon$ is a mollification of ${{\mathcal W}}$ and $(c^\varepsilon)_{\varepsilon>0}$ is a sequence of diverging constants, may converge to different limits satisfying different boundary conditions as $\varepsilon \to 0$, depending on which mollifier was used for ${{\mathcal W}}^\varepsilon$. But if the noise is only mollified in space and white in time, then the limit is always the same and it agrees with the Cole-Hopf solution with Neumann boundary conditions. Since our results show that the Cole-Hopf solution exhibits “non-physical” boundary behavior, this suggests that in order to obtain the “correct” limit it is not only necessary to subtract a large diverging constant $c^\varepsilon$ in , but additionally one should perform a boundary renormalization. Indeed, under boundary conditions the solution ${{\mathcal Z}}^\varepsilon$ to  is not spatially homogeneous, and therefore there is no reason to renormalize it by subtracting a constant $c^\varepsilon$.

Outline of the paper:
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In Section \[s2\] we give the precise definition of our microscopic dynamics and its invariant measures, we also introduce all the spaces of test 