## Journal of the Mathematical Society of Japan

### 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 1 (2013), 13-35.

Dates
First available in Project Euclid: 24 January 2013

https://projecteuclid.org/euclid.jmsj/1359036447

Digital Object Identifier
doi:10.2969/jmsj/06510013

Mathematical Reviews number (MathSciNet)
MR3034397

Zentralblatt MATH identifier
1274.60214

#### Citation

OH, Tadahiro; QUASTEL, Jeremy. On invariant Gibbs measures conditioned on mass and momentum. J. Math. Soc. Japan 65 (2013), no. 1, 13--35. doi:10.2969/jmsj/06510013. https://projecteuclid.org/euclid.jmsj/1359036447

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