An effective field theory during inflation II: stochastic dynamics and
  power spectrum suppression
  We obtain the non-equilibrium effective action of an inflaton like scalar
field --the system-- by tracing over sub Hubble degrees of freedom of
"environmental" light scalar fields. The effective action is stochastic leading
to effective Langevin equations of motion for the fluctuations of the
inflaton-like field, with self-energy corrections and stochastic noise
correlators that obey a de Sitter space-time analog of a fluctuation
dissipation relation. We solve the Langevin equation implementing a dynamical
renormalization group resummation of the leading secular terms and obtain the
corrections to the power spectrum of super Hubble fluctuations of the inflaton
field, $\mathcal{P}(k;\eta) = \mathcal{P}_0(k)\,e^{-\gamma(k;\eta)}$ where
$\mathcal{P}_0(k)$ is the nearly scale invariant power spectrum in absence of
coupling. $\gamma(k;\eta)>0$ describes the suppression of the power spectrum,
it features Sudakov-type double logarithms and entails violations of scale
invariance. We also obtain the effective action for the case of a heavy scalar
field of mass $ M \gg H$, this case yields a local "Fermi" limit with a very
weak self-interaction of the inflaton-like field and dissipative terms that are
suppressed by powers of $H/M$. We conjecture on the possibility that the large
scale anomalies in the CMB may originate in dissipative processes from inflaton
coupling to sub-Hubble degrees of freedom.
