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As indicated before, it is interesting to connect real-space and pseudo-spectral discretiza- tion approaches. From Eq. (32) [with h ( ja, t ) ≡ h$_{j}$ ( t )] we have 

and 

By equating this expression to 

we might think of the pseudo-spectral discretization as a particular real-space discretization, whose coefficients are the solutions of the linear system 

This equation is linear and homogeneous, and so it admits the trivial solution b$_{l}$ = 0 ∀ l ∈ { 1 , · · · , M } . This equation expresses in fact the fundamental difference of the spectral and finite differences discretization: the lattice spectrum. For the finite difference scheme the discrete Laplacian is no longer (2 πk/L ) 2 (namely twice the spectrum in the continuum) but 1 − cos 2 πkx L . For x = L/ 2, already for k = 1 the difference is π $^{2}$/ 2 − 2 ≈ 3. If we equate instead Eq. (39) to 

then Eq. (40) says nothing new. There is still a complete arbitrariness in the choice of the coefficients b$_{l}$ . This corresponds to the fact that this scheme is nothing but the Fourier transformed version of the finite differences one. 

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## V. GALILEAN INVARIANCE AND FLUCTUATION–DISSIPATION RELATION 

There are two main symmetries associated with the 1D KPZ equation: the fluctuation– dissipation relation and Galilean invariance. On one hand the fluctuation–dissipation rela- tion essentially tells us that the nonlinearity is asymptotically (that is, at long times) not operative in 1D. On the other hand, Galilean invariance has been traditionally related to the exact relation among exponents α + z = 2, that holds for all spatial dimensions [22, 23]. However, it is worth remarking that this interpretation has been recently criticized [24]. 

