Gibbs dynamics for fractional nonlinear Schrödinger equations with weak dispersion

We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear fractional Schr{ö}dinger equation (FNLS) with initial data distributed via its associated Gibbs measure. We construct global strong solutions with the flow property for the 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.