Gibbs measure dynamics for the fractional NLS

We construct global solutions on a full measure set with respect to the Gibbs measure for the one dimensional cubic fractional nonlinear Schr\"odinger equation (FNLS) with weak dispersion $(-\partial_x^2)^{\alpha/2}$, $\alpha<2$ by quite different methods, depending on the value of $\alpha$. We show that if $\alpha>\frac{6}{5}$, the sequence of smooth solutions for FNLS with truncated initial data converges almost surely, and the obtained limit has recurrence properties as the time goes to infinity. The analysis requires to go beyond the available deterministic theory of the equation. When $1<\alpha\leq \frac{6}{5}$, we are not able so far to get the recurrence properties but we succeeded to use a method of Bourgain-Bulut to prove the convergence of the solutions of the FNLS equation with regularized both data and nonlinearity. Finally, if $\frac{7}{8}<\alpha\leq 1$ we can construct global solutions in a much weaker sense by a classical compactness argument.