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

Abstract

We consider the Cauchy problem for the fractional nonlinear Schr\"{o}dinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter $\alpha > 1$, subject to a Gaussian random initial data of negative Sobolev regularity $\sigma<s-\frac{1}{2}$, for $s \le \frac 12$. We show that for all $s_{*}(\alpha) <s\leq \frac{1}{2}$, the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on $H^{s}(\mathbb T)$ is quasi-invariant under the flow of the equation. For $\alpha < \frac{1}{20}(17 + 3\sqrt{21}) \approx 1.537$, the regularity of the initial data is lower than the one provided by the deterministic well-posedness theory. We obtain this result by following the approach of DiPerna-Lions (1989); first showing global-in-time bounds for the solution of the infinite-dimensional Liouville equation for the transport of the Gaussian measure, and then transferring these bounds to the solution of the equation by adapting 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.