  We discuss the effective infrared theory governing a light scalar's long
wavelength dynamics in de Sitter spacetime. We show how the separation of
scales around the physical curvature radius $k/a \sim H$ can be performed
consistently with a window function and how short wavelengths can be integrated
out in the Schwinger-Keldysh path integral formalism. At leading order, and for
time scales $\Delta t \gg H^{-1}$, this results in the well-known Starobinsky
stochastic evolution. However, our approach allows for the computation of
quantum UV corrections, generating an effective potential on which the
stochastic dynamics takes place. The long wavelength stochastic dynamical
equations are now second order in time, incorporating temporal scales $\Delta t
\sim H^{-1}$ and resulting in a Kramers equation for the probability
distribution - more precisely the Wigner function - in contrast to the more
usual Fokker-Planck equation. This feature allows us to non-perturbatively
evaluate, within the stochastic formalism, not only expectation values of field
correlators, but also the stress-energy tensor of $\phi$.
