VIII. CONCLUSIONS The moral from the present analysis is clear: due to the constraint imposed by the Hopf– Cole transformation [Eqs. (3) and (6)] the discrete forms of the Laplacian and the nonlinear KPZ term cannot be chosen independently. Although the present work is focused on the relation between the diffusion equation with multiplicative noise and the KPZ equation, the consequences of this analysis are more general. The discrete versions of any set of related differential equations should be obtained taking into account the original (or leading) equation and the transformation rules. It is worth remarking here that a related analysis was done in [9], but there the emphasis was on the study of the strong coupling limit and the mapping onto the directed polymer problem, without commenting at all about the consistency among the discrete versions of the differential operators. The results discussed here are general; they neither depend on space dimensionality nor rely on variational representations. Nonetheless, the recently introduced variational ap- proach for KPZ [26] offers an adequate framework in order to make a consistent discretization of the KPZ equation. Regarding the recently introduced pseudo-spectral approach [14, 16, 36], in addition to its known advantages, in principle, it seems to have the virtue of being “transparent” respect to the present problem. In this respect, we have shown the relation that exists between it and the present analysis. Besides that, when analyzing inhomogeneous situations where defects or impurities are present, such methods do not apply and it is again necessary to resort to real-space discrete form of the differential operators [21, 37]. Another aspect to consider is related to the situation in [33], where a problem of numerical instabilities (a computational problem) in discrete growth models has been tackled by introducing higher order contributions (changing the physics of the problem!). It is worth indicating that such an instability does not seem to arise (or at least it arises latter) in pseudo-spectral treatments of the same problem. Hence, due to the relation among both formalisms, it seems reasonable to expect that such instabilities could at least be delayed if a consistent discretization scheme, together with higher order discrete operators, is used. Regarding the two main symmetries associated with the 1D KPZ equation, the fluctuation–dissipation relation and Galilean invariance, we have found a couple of relevant 20