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

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 $L^2$.