Invariant measures for the periodic derivative nonlinear Schr\"odinger equation

TL;DR

This paper constructs invariant measures for the periodic derivative nonlinear Schrödinger equation (DNLS) for small initial data, showing these measures are absolutely continuous relative to Gaussian measures, advancing understanding of DNLS dynamics.

Contribution

It introduces a novel method to construct invariant measures for DNLS and analyzes the gauge group, establishing absolute continuity with Gaussian measures for small data.

Findings

Invariant measures are constructed for small $L^2$ data.

Measures are shown to be absolutely continuous with respect to Gaussian measures.

Quasi-invariance of Gaussian measures under gauge transformations is proved.

Abstract

We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation (DNLS) for small data in $L^2$ and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of Gaussian measures on $L^2$.