Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem for the fractional nonlinear Schrödinger 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 $\sigma <s-1/2$, for $s\le 1/2$. We show that for all $s_{\ast }(\alpha )<s\le 1/2$, 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. This is the first probabilistic globalization argument that is not in the setting of an invariant measure and not based on a known deterministic method to construct global-in-time solutions. 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 the probabilistic local well-posedness theory.