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January, 2013 On invariant Gibbs measures conditioned on mass and momentum
Tadahiro OH, Jeremy QUASTEL
J. Math. Soc. Japan 65(1): 13-35 (January, 2013). DOI: 10.2969/jmsj/06510013

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.

Citation

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Tadahiro OH. Jeremy QUASTEL. "On invariant Gibbs measures conditioned on mass and momentum." J. Math. Soc. Japan 65 (1) 13 - 35, January, 2013. https://doi.org/10.2969/jmsj/06510013

Information

Published: January, 2013
First available in Project Euclid: 24 January 2013

zbMATH: 1274.60214
MathSciNet: MR3034397
Digital Object Identifier: 10.2969/jmsj/06510013

Subjects:
Primary: 60H40
Secondary: 35Q53, 35Q55, 60H30

Rights: Copyright © 2013 Mathematical Society of Japan

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Vol.65 • No. 1 • January, 2013
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