Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations
Justin Forlano
University of Edinburgh, Edinburgh, UKLeonardo Tolomeo
University of Edinburgh, Edinburgh, UK

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 , subject to a Gaussian random initial data of negative Sobolev regularity , for . We show that for all , the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on is quasi-invariant under the flow of the equation. For , 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.
Cite this article
Justin Forlano, Leonardo Tolomeo, Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations. J. Eur. Math. Soc. (2025), published online first
DOI 10.4171/JEMS/1643