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

  • Justin Forlano

    University of Edinburgh, Edinburgh, UK
  • Leonardo Tolomeo

    University of Edinburgh, Edinburgh, UK
Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations cover
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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