assuming that the initial condition is h$_{j}$ (0) = 0. Applying Jensen’s inequality to this last relation one finds in agreement with what one could directly obtain from the KPZ equation [Eq. (1)]. An important feature of the Hopf–Cole transformation—Eq. (3) or (6)—is that it is local , i.e. it involves neither spatial nor temporal transformations. Some effects of this feature are the following In this section, we want to go further with the criterion that the definitions of the discrete operators should not depend on the fields on which they are applied. The deterministic part of Eq. (2), namely the diffusion term, admits a local Lyapunov functional. In other words, for ε = 0, Eq. (2) can be written in the following variational form with The aforementioned criterion dictates the following set of discrete forms (thus Lyapunov functions, for any finite N ) of Eq. (20) It is a trivial task to verify that 8 ν L$_{(}$$_{n}$$_{)}$ ( φ$_{j}$ ) = − 1 a ∂ F$_{(}$$_{n}$$_{)}$ [ φ ] ∂φ$_{j}$ . (22) F$_{(}$$_{n}$$_{)}$ [ φ ] = 1 2 νa N ∑ j =1 Q$_{(}$$_{n}$$_{)}$ ( φ$_{j}$ ) . (21) F [ φ ] = ν 2 ∫ d x ( ∂$_{x}$φ ) $^{2}$. (20) ∂$_{t}$φ = − δ F [ φ ] δφ , (19) III. EXPLOITING THE DETERMINISTIC LYAPUNOV FUNCTIONAL 1. The discrete form of the Laplacian—namely the operator L$_{(}$$_{n}$$_{)}$ —is the same, regardless of whether it is applied to φ or to h . 2. For a given L$_{(}$$_{n}$$_{)}$ (i.e. a given set of b$_{l}$ = 0 , l = 1 . . . n ), Q$_{(}$$_{n}$$_{)}$ is determined by the Hopf–Cole transformation, Eq. (6). 〈 h$_{j}$ ( t ) 〉 ≥ F$_{eff}$ t, (18)