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

Abstract

We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger 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}(\mathbb T) with s \ge \frac{1}{2} , 2 < r < 4 , (s-1)r <-1 and scaling like H^{\frac{1}{2}-\epsilon}(\mathbb T), for small \epsilon >0 . We also show the invariance of this measure.