II. THE LAPLACIAN DETERMINES THE NONLINEAR TERM In this section we elucidate—by considering the standard  nearest-neighbor discretization prescription as a benchmark—one of two constraints to be obeyed by any spatial discretiza- tion scheme. It is very important to remark that this constraint arises due to the mapping between the KPZ and the diffusion equation (with multiplicative noise) through the Hopf– Cole transformation. Hence  for a general real-space discrete Laplacian  we state the form of its corresponding KPZ term. Even though the present analysis is performed on the KPZ equation  it is general in the sense that for sets of equations related among themselves through a local transformation there should be a consistent relation between the discrete transformed forms. A. The simplest case As it is known  the diffusion equation with multiplicative noise ∂$_{t}$φ = ν ∂ 2 $_{x}$φ + λF 2 ν φ + λε 2 ν φ ξ  (2) is related to the KPZ equation [Eq. (1)] through the Hopf–Cole transformation φ ( x  t ) = exp [ λ 2 ν h ( x  t ) ] . (3) Note that this transformation is just one particular example of the general implicit trans- formation written down in Ref. [28]. The standard spatial discrete version of Eq. (2)  after transforming to a co-moving refer- ence frame φ → φ + F t   is ˙ φ$_{j}$ = ν L$_{(1)}$ ( φ$_{j}$ ) + λε 2 ν φ$_{j}$ξ$_{j}$  (4) with 1 ≤ j ≤ N ≡ 0  because periodic boundary conditions are assumed as usual (the implicit sum convention is not meant in any of the discrete expressions). The discrete noise ξ$_{j}$ ( t ) is a Gaussian random variable with zero mean and correlation given by 〈 ξ$_{j}$ ( t ) ξ$_{k}$ ( t ) 〉 = 2 δ$_{jk}$ a δ ( t − t $^{′}$) . (5) Then  using the discrete version of Eq. (3) φ$_{j}$ ( t ) = exp [ λ 2 ν h$_{j}$ ( t ) ]   (6) 5