A. Galilean Invariance Galilean invariance means that the KPZ equation is invariant under the transformation where v is an arbitrary constant vector field and F is the external (constant) driving force. Using the classical discretization for the (complete) KPZ equation, we find One can immediately check that this equation is invariant under the discrete Galilean trans- formation However, Eq. (43) has been criticized for its instability properties, at least when the spatial discretization is not fine enough [9]. If we use the alternative discretization we find that this equation is not invariant under the discrete Galilean transformation. In fact, the transformation h → h + vja yields an excess term which is compatible with the gradient discretization in Eq. (42); however, this discretization does not allow to recover the quadratic term in Eq. (45), indicating that this finite differences scheme does not fulfill Galilean invariance. The Hopf–Cole transformed equation 13 ˙ φ$_{j}$ = νL$_{j}$ ( φ ) + λF 2 ν φ$_{j}$ + λ 2 ν φ$_{j}$ ξ$_{j}$ ( t ) , (46) ˙ h$_{j}$ = νL$_{1}$ ( h$_{j}$ ) + λ 4 [( H j +1 j ) 2 + ( H j j − 1 ) 2 ] + F + ξ$_{j}$ ( t ) , (45) j a → j a − λv t, h$_{j}$ → h$_{j}$ + v j a, F → F − λ 2 v $^{2}$. (44) ˙ h$_{j}$ = ν L$_{1}$ + λ 8 ( H j +1 j − 1 ) 2 + F + ξ$_{j}$ ( t ) . (43) ∂$_{x}$h → 1 2 H j +1 j − $_{1}$, (42) x → x − λv t, h → h + v x, F → F − λ 2 v $^{2}$, (41)