Invariant measure construction at a fixed mass

TL;DR

This paper constructs an invariant measure for the derivative nonlinear Schrödinger equation on the torus at fixed small mass using a divergence theorem analogy in infinite dimensions, and proves the integrability of the measure's density.

Contribution

It introduces a novel method to construct invariant measures at fixed mass for the derivative NLS using an infinite-dimensional divergence theorem approach.

Findings

Invariant measure constructed at small fixed mass.

Density function of the measure is in L^p after Fourier scaling.

Method emulates divergence theorem in infinite dimensions.

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.