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

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