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

• Published in: Journal of the European Mathematical Society : JEMS
• Main Authors: Forlano, Justin, Tolomeo, Leonardo
• Format: Journal Article
• Language: English
• Published: 03.06.2025
• ISSN: 1435-9855; 1435-9863; 1435-9863
• DOI: 10.4171/jems/1643

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\leq{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.