On invariant Gibbs measures conditioned on mass and momentum

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We construct a Gibbs measure for the nonlinear Schrödinger equation (NLS) on the circle, conditioned on prescribed mass and momentum:<br>$$d \mu_{a,b} = Z^{-1} 1_{ \{\int_{\mathbb{T}} |u|^2 = a\}} 1_{\{i \int_{\mathbb{T}} u \overline{u}_x = b\}}e^{\pm 1/p \int_{\mathbb{T}} |u|^p-1/2\int_{\mathbb{T}} |u|^2 } d P$$<br>for a ∈ ℝ+ and b ∈ ℝ, where P is the complex-valued Wiener measure on the circle. We also show that μa,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.

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