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

### Andrea R. Nahmod

University of Massachusetts, Amherst, USA### Tadahiro Oh

Princeton University, USA### Luc Rey-Bellet

University of Massachusetts, Amherst, USA### Gigliola Staffilani

Massachusetts Institute of Technology, Cambridge, USA

## 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 ${\mathcal F}L^{s,r}(\T)$ 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.

## Cite this article

Andrea R. Nahmod, Tadahiro Oh, Luc Rey-Bellet, Gigliola Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS. J. Eur. Math. Soc. 14 (2012), no. 4, pp. 1275–1330

DOI 10.4171/JEMS/333