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We consider the Cauchy problem for the fractional nonlinear Schr\"{o}dingerequation (FNLS) on the one-dimensional torus with cubic nonlinearity and highdispersion parameter $\alpha > 1$, subject to a Gaussian random initial data ofnegative Sobolev regularity $\sigma<s-\frac{1}{2}$, for $s \le \frac 12$. Weshow that for all $s_{*}(\alpha) <s\leq \frac{1}{2}$, the equation is almostsurely globally well-posed. Moreover, the associated Gaussian measure supportedon $H^{s}(\mathbb T)$ is quasi-invariant under the flow of the equation. For$\alpha < \frac{1}{20}(17 + 3\sqrt{21}) \approx 1.537$, the regularity of theinitial data is lower than the one provided by the deterministic well-posednesstheory.  We obtain this result by following the approach of DiPerna-Lions (1989);first showing global-in-time bounds for the solution of theinfinite-dimensional Liouville equation for the transport of the Gaussianmeasure, and then transferring these bounds to the solution of the equation byadapting Bourgain's invariant measure argument to the quasi-invariance setting.This allows us to bootstrap almost sure global bounds for the solution of(FNLS) from its probabilistic local well-posedness theory.

Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schr\"odinger equations