Gibbs measure dynamics for the fractional NLS

Understand

We construct global solutions on a full measure set with respect to the Gibbs measure for the one dimensional cubic fractional nonlinear Schr\&#34;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.