Invariant measure construction at a fixed mass

Abstract

In this paper we analyze the derivative nonlinear Schr\"odinger 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.