  In this work we generalize a previously developed semiclassical approach to
inflation, devoted to the analysis of the effective dynamics of coarse-grained
fields, which are essential to the stochastic approach to inflation. We
consider general non-trivial momentum distributions when defining these fields.
The use of smooth cutoffs in momentum space avoids highly singular quantum
noise correlations and allows us to consider the whole quantum noise sector
when analyzing the conditions for the validity of an effective classical
dynamical description of the coarse-grained field. We show that the weighting
of modes has physical consequences, and thus cannot be considered as a mere
mathematical artifact. In particular we discuss the exponential inflationary
scenario and show that colored noises appear with cutoff dependent amplitudes.
