Gibbs measure for the periodic derivative nonlinear Schr\"odinger equation

TL;DR

This paper constructs a Gibbs measure for the periodic derivative nonlinear Schrödinger equation using renormalization techniques and Wiener chaos estimates, extending methods previously applied to other nonlinear dispersive equations.

Contribution

It introduces a novel construction of Gibbs measures for the derivative NLS on the circle, employing renormalization and Wiener chaos methods.

Findings

Successful construction of the Gibbs measure for the derivative NLS

Application of renormalization and Wiener chaos techniques

Extension of methods from Benjamin-Ono to derivative NLS

Abstract

In this paper we construct a Gibbs measure for the derivative Schr\"odinger equation on the circle. The construction uses some renormalisations of Gaussian series and Wiener chaos estimates, ideas which have already been used by the second author in a work on the Benjamin-Ono equation.