  Combining the stochastic and $\delta N$ formalisms, we derive non
perturbative analytical expressions for all correlation functions of scalar
perturbations in single-field, slow-roll inflation. The standard, classical
formulas are recovered as saddle-point limits of the full results. This yields
a classicality criterion that shows that stochastic effects are small only if
the potential is sub-Planckian and not too flat. The saddle-point approximation
also provides an expansion scheme for calculating stochastic corrections to
observable quantities perturbatively in this regime. In the opposite regime, we
show that a strong suppression in the power spectrum is generically obtained,
and comment on the physical implications of this effect.
