  We consider the Cauchy problem for the one-dimensional periodic cubic
nonlinear fractional Schr\"odinger equation (FNLS) with initial data
distributed via Gibbs measure. We construct global strong solutions with the
flow property to FNLS on the support of the Gibbs measure in the full
dispersive range, thus resolving a question proposed by Sun-Tzvetkov (2021). As
a byproduct, we prove the invariance of the Gibbs measure and almost sure
global well-posedness for FNLS.
