Asymptotic stability of a wide class of steady states for the Hartree equation for random fields

We study the Hartree equation describing the time evolution of the wave functions of infinitely many fermions interacting with each other. The Hartree equation can be formulated in terms of random fields. This formulation was introduced by de Suzzoni in [arXiv:1507.06180]. It has infinitely many steady states, and [arXiv:1811.03150, arXiv:2007.00472] have studied its asymptotic stability. However, they required somewhat strong conditions for steady states, and the stability of Fermi gas at zero temperature, one of the most important steady states from the physics point of view, was left open. In this paper, we prove the stability of steady states in a wide class, which includes Fermi gas at zero temperature in $d$-dimensional space when $d \ge 3$. This paper is the revised version. There are two major changes. The first one is that the assumptions for potential $w$ in Theorem 1.3 are weakened. The second one is that we removed Theorem 1.4 in the previous version because the author found some gaps which seemed difficult to fix.