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We establish higher order convergence rates in periodic homogenization offully nonlinear uniformly parabolic Cauchy problems accompanied with rapidlyoscillating initial data. Such result is new even for linear problems. Here weconstruct higher order initial layer and interior correctors, which describethe oscillatory behavior near the initial and interior time zone of the domain.To construct higher order correctors, we develop a regularity theory inmacroscopic scales, and prove an exponential decay estimate for initial layercorrectors.  The higher order expansion requires an iteration process: successivelycorrecting the initial layer, then the interior. This leads to a morecomplicated 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 givenoperator is space-time periodic, the interior correctors become aperiodic inthe time variable as we proceed with the iteration process. Moreover, eachinterior corrector of higher order is paired with a space-time periodicversion, and the difference between the two decays exponentially fast withtime.

Higher Order Convergence Rates in Theory of Homogenization II: Oscillatory Initial Data