In this paper we consider the defocusing Hartree nonlinear Schr\"odinger equations on $\mathbb T^3$ with real valued and even potential $V$ and Fourier multiplier decaying like $|k|^{ -\beta}$. By relying on the method of random averaging 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 the associated Gibbs measure and global existence of strong solutions in its statistical ensemble. In this way we extend Bourgain's seminal result [7] which requires $\beta >2$ in this case.
Categories: math.AP