I. INTRODUCTION Soon after its formulation in 1986, the KPZ equation [1–3] became a paradigm as the description of a vast class of nonequilibrium phenomena by means of stochastic fields. The field h ( x, t ) whose evolution is governed by this stochastic nonlinear partial differential equation, describes the height of a fluctuating interface in the context of surface growth processes in which it was originally formulated. In particular, Eq. (1) assumes a one-dimensional (1D) homogeneous substrate of size L . The parameter ν determines the surface tension, λ is proportional to the average growth velocity (the surface slope is parallel- transported in the growth process), and F is an external driving force. Finally, ξ ( x, t ) is a Gaussian white noise with 〈 ξ ( x, t ) 〉 = 0 and 〈 ξ ( x, t ) ξ ( x $^{′}$, t$^{′}$) 〉 = 2 δ ( x − x $^{′}$) δ ( t − t $^{′}$). As usual, periodic boundary conditions are assumed. From a theoretical point of view the KPZ equation has many interesting properties, like its close relationship with the Burgers equation [4] or with a diffusion equation with multiplicative noise, whose field φ ( x, t ) can be interpreted as the restricted partition function of the directed polymer problem. But clearly, investigating the behavior of its solutions to obtain e.g. the critical exponents in one or more spatial dimensions [5–13] requires the (stochastic) numerical integration of a discrete version. Although a pseudo-spectral spatial discretization scheme has been recently put forward [14–16], as well as a numerical large deviation theory [4], real-space discrete versions of Eq. (1) are still largely used for numerical simulations [17–20], because of their relative ease of implementation and of interpretation in the case of non-homogeneous substrates (for instance, a quenched impurity distribution [21]) among other reasons. To that end, several real-space discretization schemes have been proposed [10, 11, 19], which are claimed to cure particular “diseases” of the numerical simulation. In the present work, no attempt is made of comparing alternative real-space discretiza- tion schemes in sought of special KPZ features. Instead, we seek to point out some basic conditions that any spatial discretization must fulfill in order to consistently describe the KPZ equation. Nonetheless, for the sake of brevity and for ease of comparison with other 2 ∂$_{t}$h = ν ∂ 2 $_{x}$h + λ 2 ( ∂$_{x}$h ) 2 + F + ε ξ ( x, t ) , (1)