Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

In this paper we construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schr\"odinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier-Lebesgue space {\mathcal F}L^{s,r}(\T) with s \geq 1/2, 2 < r, <4, (s-1)r < -1, and scaling like H^{\frac{1}{2}-\epsilon}(\T), for small \epsilon > 0.  We also show the invariance of this measure.