  Langevin equations for several competitive growth models in one dimension are
derived. For models with crossover from random deposition (RD) to some
correlated deposition (CD) dynamics, with small probability p of CD, the
surface tension \nu and the nonlinear coefficient \lambda of the associated
equations have linear dependence on p due solely to this random choice.
However, they also depend on the regularized step functions present in the
analytical representations of the CD, whose expansion coefficients scale with p
according to the divergence of local height differences when p->0. The
superposition of those scaling factors gives \nu ~ p^2 for random deposition
with surface relaxation (RDSR) as the CD, and \nu ~ p, \lambda ~ p^{3/2} for
ballistic deposition (BD) as the CD, in agreement with simulation and other
scaling approaches. For bidisperse ballistic deposition (BBD), the same scaling
of RD-BD model is found. The Langevin equation for the model with competing
RDSR and BD, with probability p for the latter, is also constructed. It shows
linear p-dependence of \lambda, while the quadratic dependence observed in
previous simulations is explained by an additional crossover before the
asymptotic regime. The results highlight the relevance of scaling of the
coefficients of step function expansions in systems with steep surfaces, which
is responsible for noninteger exponents in some p-dependent stochastic
equations, and the importance of the physical correspondence of aggregation
rules and equation coefficients.
