A Scattering Result Around a Nonlocalized Equilibria for the Quintic Hartree Equation for Random Fields

We consider a quintic Hartree equation for a random field, which describes the temporal evolution of an infinitely many particles, considering a three-body interaction. We show a scattering result around a nonlocalized equilibria of the equation, for high dimensions \(d\ge 4\) $d\ge 4$ . The Hartree equation for random variables was introduced by de Suzzoni (in: Séminaire Laurent Schwartz—EDP et applications (2015–2016) (fr), talk:12) but only for a two-body interaction, that leads to a cubic Hartree equation for random variables. Scattering results for the cubic Hartree equation have been shown in Collot and de Suzzoni (J Math Pures Appl 137(9):70–100, 2020; Ann Henri Lebesgue 2022), and we extend those results to the quintic Hartree equation. As in Collot and de Suzzoni (Ann Henri Lebesgue 2022) we consider a large range of potentials that includes the Dirac delta