  We extend the `moment transport method' for calculating the statistics of
inflationary perturbations to the quantum phase of evolution on sub-horizon
scales. The quantum transport equations form a set of coupled ordinary
differential equations for the evolution of quantum correlation functions
during inflation, which are valid on sub- and super-horizon scales, and reduce
to the known classical transport equations after horizon crossing. The
classical and quantum equations follow directly from the field equations of
cosmological perturbation theory. In this paper, we focus on how the evolution
equations arise, and explore how transport methods relate to other approaches,
and in particular how formal integral solutions to the transport equations
connect to those of the In-In formalism.
