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 $H^{\frac{1}{2}-ϵ}(\mathbb{T}),$ for small $ϵ>0$. We also show the invariance of this measure.