  We consider a Hartree equation for a random variable, which describes the
temporal evolution of infinitely many Fermions. On the Euclidean space, this
equation possesses equilibria which are not localised. We show their stability
through a scattering result, with respect to localised perturbations in the
defocusing case in high dimensions $d\geq 4$. This provides an analogue of the
results of Lewin and Sabin \cite{LS2}, and of Chen, Hong and Pavlovi\'c
\cite{CHP2} for the Hartree equation on operators. The proof relies on
dispersive techniques used for the study of scattering for the nonlinear
Schr\"odinger and Gross-Pitaevskii equations.
