On invariant Gibbs measures conditioned on mass and momentum

TL;DR

This paper constructs a Gibbs measure for the nonlinear Schrödinger equation on the circle conditioned on fixed mass and momentum, and proves its invariance under the NLS flow, highlighting the role of the Levy stochastic area.

Contribution

It introduces a new Gibbs measure conditioned on mass and momentum for NLS and proves its invariance under the flow, incorporating the Levy stochastic area.

Findings

Gibbs measure conditioned on mass and momentum constructed

Invariance of the measure under NLS flow proved

Connection between Levy stochastic area and invariance

Abstract

We construct a Gibbs measure for the nonlinear Schrodinger equation (NLS) on the circle, conditioned on prescribed mass and momentum: d \mu_{a,b} = Z^{-1} 1_{\int_T |u|^2 = a} 1_{i \int_T u \bar{u}_x = b} exp (\pm1/p \int_T |u|^p - 1/2 \int_{\T} |u|^2) d P for a \in R^+ and b \in R, where P is the complex-valued Wiener measure on the circle. We also show that \mu_{a,b} is invariant under the flow of NLS. We note that i \int_\T u \bar{u}_x is the Levy stochastic area, and in particular that this is invariant under the flow of NLS.