Absolute continuity of Brownian bridges under certain gauge transformations

TL;DR

This paper proves the absolute continuity of Gaussian measures related to complex Brownian bridges under specific gauge transformations and demonstrates the invariance of a particular measure for the periodic derivative nonlinear Schrödinger equation.

Contribution

It establishes the absolute continuity of Gaussian measures under gauge transformations and confirms the invariance of a measure for the derivative nonlinear Schrödinger equation.

Findings

Gaussian measures are absolutely continuous under certain gauge transformations

The invariant measure for the periodic derivative NLS coincides with a weighted Wiener measure

The measure constructed by Thomann and Tzvetkov is invariant under the flow

Abstract

We prove absolute continuity of Gaussian measures associated to complex Brownian bridges under certain gauge transformations. As an application we prove that the invariant measure for the periodic derivative nonlinear Schr\"odinger equation obtained by Nahmod, Oh, Rey-Bellet and Staffilani in [20], and with respect to which they proved almost surely global well-posedness, coincides with the weighted Wiener measure constructed by Thomann and Tzvetkov [24]. Thus, in particular we prove the invariance of the measure constructed in [24].