  Our previous work [37] presented a rigorous derivation of quantum Boltzmann
equations near a Bose-Einstein condensate (BEC). Here, we extend it with a
complete characterization of the leading order fluctuation dynamics. For this
purpose, we correct the latter via an appropriate Bogoliubov rotation, in
partial analogy to the approach by Grillakis-Machedon et al. [59], in addition
to the Weyl transformation applied in [37]. Based on the analysis of the third
order expansion of the BEC wave function, and the second order expansions of
the pair-correlations, we show that through a renormalization strategy, various
contributions to the effective Hamiltonian can be iteratively eliminated by an
appropriate choice of the Weyl and Bogoliubov transformations. This leads to a
separation of renormalized Hartree-Fock-Bogoliubov (HFB) equations and quantum
Boltzmann equations. A multitude of terms that were included in the error term
in [37] are now identified as contributions to the HFB renormalization terms.
Thereby, the error bound in the work at hand is improved significantly. To the
given order, it is now sharp, and matches the order or magnitude expected from
scaling considerations. Consequently, we extend the time of validity to $t\sim
(\log N)^2$ compared to $t\sim (\log N/\log \log N)^2$ before. We expect our
approach to be extensible to smaller orders in $\frac1N$.
