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

In this paper, we consider the defocusing Hartree nonlinear Schrödinger equations on T3 with real-valued and even potential V and Fourier multiplier decaying such as |k|−β. By relying on the method of random averaging operators [Deng et al., arXiv:1910.08492 (2019)], we show that there exists β0, which is less than but close to 1, such that for β > β0, 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 [J. Bourgain, J. Math. Pures Appl. 76, 649–702 (1997)], which requires β > 2 in this case.