  We study the Gross-Pitaevskii hierarchy on the spatial domain $\mathbb{T}^3$.
By using an appropriate randomization of the Fourier coefficients in the
collision operator, we prove an averaged form of the main estimate which is
used in order to contract the Duhamel terms that occur in the study of the
hierarchy. In the averaged estimate, we do not need to integrate in the time
variable. An averaged spacetime estimate for this range of regularity exponents
then follows as a direct corollary. The range of regularity exponents that we
obtain is $\alpha>\frac{3}{4}$. It was shown in our previous joint work with
Gressman that the range $\alpha>1$ is sharp in the corresponding deterministic
spacetime estimate. This is in contrast to the non-periodic setting, which was
studied by Klainerman and Machedon, in which the spacetime estimate is known to
hold whenever $\alpha \geq 1$. The goal of our paper is to extend the range of
$\alpha$ in this class of estimates in a \emph{probabilistic sense}.
  We use the new estimate and the ideas from its proof in order to study
randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider
hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the
collision operator has been randomized. For these hierarchies, we show
convergence to zero in low regularity Sobolev spaces of Duhamel expansions of
fixed deterministic density matrices. We believe that the study of the
randomized collision operators could be the first step in the understanding of
a nonlinear form of randomization.
