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In this paper we consider the defocusing Hartree nonlinear Schr\"odingerequations on $\mathbb T^3$ with real valued and even potential $V$ and Fouriermultiplier decaying like $|k|^{-\beta}$. By relying on the method of randomaveraging operators in arXiv:1910.08492, we show that there exists $\frac{1}{2}\ll \beta_0 <1 $ such that for $ \beta > \beta_0 $ we have invariance of theassociated Gibbs measure and global existence of strong solutions in itsstatistical ensemble. In this way we extend Bourgain's seminal result [7] whichrequires $\beta >2$ in this case.

Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three