Invariant measure construction at a fixed mass

Abstract

In this paper we analyze the derivative nonlinear Schrödinger equation on $\mathbb{T}$ with randomized initial data in ${\cap }_{s<\frac{1}{2}}{H}^s(\mathbb{T})$ according to a Wiener measure. We construct an invariant measure at each sufficiently small, fixed mass $m$ through an argument that emulates the divergence theorem in infinitely many dimensions. We also prove that the density function needed to construct the Wiener measure is in $L^p$, even after scaling of the Fourier coefficients of the intial data.

A previous version contained a serious error in Lemma 5.9. This is the correct version

References in corpus (1)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data

Cited by in corpus (2)

• Invariance of white noise for KdV on the line
• Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus