Journal of the Mathematical Society of Japan

On invariant Gibbs measures conditioned on mass and momentum

Tadahiro OH and Jeremy QUASTEL

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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

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

First available in Project Euclid: 24 January 2013

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Zentralblatt MATH identifier

Primary: 60H40: White noise theory
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Gibbs measure Schrödinger equation Kortweg-de Vries equation Lévy area


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.

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