Almost sure scattering for defocusing energy critical Hartree equation on ${\mathbb{R}}^5$

We consider the defocusing energy-critical Hartree equation $i{\text{pa}}_{t}u+\mathrm{\Delta }u=(\mathrm{\mid }\cdot {\mathrm{\mid }}^{-4}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u$ in spatial dimension $d=5$ and prove almost sure scattering with initial data $u_0\in H_x^s({\mathbb{R}}^5)$ for any $s\in \mathbb{R}$. The proof relies on the modified interaction Morawetz estimate, the stability theories, the ``Narrowed’’ Wiener randomization. We are inspired to consider this problem by the work of Shen-Soffer-Wu \cite{Shen-Soffer-Wu 1}, which treated the analogous problem for the energy-critical Schrödinger equation. The new ingredient in this paper are that we take an alternative proof to give the interaction Morawetz estimate. And the nonlocal nonlinearity term will bring some difficulties.