We consider a Hartree equation for a random field, which describes the temporal evolution of infinitely many fermions. On the Euclidean space, this equation possesses equilibria which are not localized. We show their stability through a scattering result, with respect to localized perturbations in the not too focusing case in high dimensions $d\ge 4$. This provides an analogue of the results of Lewin and Sabin [22], and of Chen, Hong and Pavlović [11] for the Hartree equation on operators. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger and Gross-Pitaevskii equations.

我们考虑一个用于随机场的Hartree方程，该方程描述了无限多个费米子的时间演化。在欧几里得空间上，该方程具有不局域的平衡点。我们通过散射结果显示了它们的稳定性，在高尺寸情况下，在不太聚焦的情况下，局部扰动$d\ge 4$。这为算子的Hartree方程提供了Lewin和Sabin [22]以及Chen，Hong和Pavlović[11]的结果的类似物。该证明依赖于用于研究非线性Schrödinger和Gross-Pitaevskii方程的散射的色散技术。