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 FL[superscript s,r](\T) with s ≥ 1/2, 2 < r < 4, (s−1)r < −1 and scaling like H1/2[superscript −ϵ](T), for small ϵ>0. We also show the invariance of this measure.