  We study loop corrections to correlation functions of inflationary
perturbations. Previous calculations have found that the two-point function can
have a logarithmic running of the form log(k/mu), where k is the wavenumber of
the perturbation, and mu is the renormalization scale. We highlight that this
result would have profound consequences for both eternal inflation and the
predictivity of standard inflation. We find a different result. We consider two
sets of theories: one where the inflaton has a large cubic self-interaction and
one where the inflaton interacts gravitationally with N massless spectator
scalar fields. We find that there is a logarithmic running but of the form
log(H/mu), where H is the Hubble constant during inflation. We find this result
in three independent ways: by performing the calculation with a sharp cutoff in
frequency-momentum space, in dimensional regularization and by the simple
procedure of making the loop integral dimensionless. For the simplest of our
theories we explicitly renormalize the correlation function proving that the
divergencies can be reabsorbed and that the correlation function for
super-horizon modes does not depend on time (once the tadpole terms have been
properly taken into account). We prove the time-independence of the
super-horizon correlation function in several additional ways: by doing the
calculation of the correlation function at finite time using both the
regularizations and by developing a formalism which expresses loop corrections
directly in terms of renormalized quantities at each time. We find this last
formalism particularly helpful to develop intuition which we then use to
generalize our results to higher loops and different interactions.
