  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.
