  We establish higher order convergence rates in periodic homogenization of
fully nonlinear uniformly parabolic Cauchy problems accompanied with rapidly
oscillating initial data. Such result is new even for linear problems. Here we
construct higher order initial layer and interior correctors, which describe
the oscillatory behavior near the initial and interior time zone of the domain.
To construct higher order correctors, we develop a regularity theory in
macroscopic scales, and prove an exponential decay estimate for initial layer
correctors.
  The higher order expansion requires an iteration process: successively
correcting the initial layer, then the interior. This leads to a more
complicated asymptotic expansion, as compared to the non-oscillating data case,
and this complexity is present even in the linear case.
  A notable observation for fully nonlinear operators is that even if the given
operator is space-time periodic, the interior correctors become aperiodic in
the time variable as we proceed with the iteration process. Moreover, each
interior corrector of higher order is paired with a space-time periodic
version, and the difference between the two decays exponentially fast with
time.
