Gibbs Dynamics for the Weakly Dispersive Nonlinear Schr\"odinger Equations

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.