  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.
