Stability of equilibria for a Hartree equation for random fields

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.