The stochastic approach to inflation relies on one key assumption, the
emergence of a long-wave classical field that drives the inflation an
d is subject to a shortwave classical noise. In this work we consider
explicitly the potential that acts on the inflaton field, and analyze
classicality conditions that must be satisfied to have an effective cl
assical stochastic approach. When these hold, the dynamics is given by
a two-dimensional classical Fokker-Planck equation. We develop some e
xamples, already widely considered in different approaches, and find a
very suggestive result, which is the damping of the quantum fluctuati
ons by the effect of the inflaton interaction potential.