On invariant Gibbs measures conditioned on mass and momentum

Abstract

We construct a Gibbs measure for the nonlinear Schrödinger equation (NLS) on the circle, conditioned on prescribed mass and momentum:

$$d \mu_{a,b} = Z^{-1} 1_{ \{\int_{\mathbb{T}} |u|^2 = a\}} 1_{\{i \int_{\mathbb{T}} u \overline{u}_x = b\}}e^{\pm 1/p \int_{\mathbb{T}} |u|^p-1/2\int_{\mathbb{T}} |u|^2 } d P$$

for $a \in \mathbb{R}^+$ and $b \in \mathbb{R}$, 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 $i \int_{\mathbb{T}} u \overline{u}_x$ is the Lévy stochastic area, and in particular that this is invariant under the flow of NLS.