Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLS

TL;DR

This paper proves that Gaussian measures with certain regularity are quasi-invariant under the gauge map associated with the periodic derivative NLS, extending previous results to non-integer regularity levels.

Contribution

It extends the quasi-invariance of Gaussian measures under the gauge map to all regularity levels $s > 1/2$, including non-integer values, for the periodic derivative NLS.

Findings

Quasi-invariance holds for all $s > 1/2$.

Extension from integer to non-integer regularity parameters.

Generalization of previous results by Nahmod et al. and Genovese et al.

Abstract

The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on $L^2(\T)$ with covariance $[1+(-\D)^{s}]^{-1}$ under these transformations for any $s>\frac12$. This extends previous achievements by Nahmod, Ray-Bellet, Sheffield and Staffilani (2011) and Genovese, Luc\`a and Valeri (2018), who proved the result for integer values of the regularity parameter $s$.