On invariant Gibbs measures conditioned on mass and momentum

Details On invariant Gibbs measures conditioned on mass and momentum On invariant Gibbs measures conditioned on mass and momentum

2010-12-15

arxiv.org/abs/1012.3432v2

Mathematics

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17 pages. An error in Subsec. 2.1 is corrected (see Prop. 2.2.) Also, an argument in Subsec. 2.3 is simplified. To appear in J

Scientific paper

We construct a Gibbs measure for the nonlinear Schrodinger equation (NLS) on the circle, conditioned on prescribed mass and momentum: d \mu_{a,b} = Z^{-1} 1_{\int_T |u|^2 = a} 1_{i \int_T u \bar{u}_x = b} exp (\pm1/p \int_T |u|^p - 1/2 \int_{\T} |u|^2) d P for a \in R^+ and b \in 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_\T u \bar{u}_x is the Levy stochastic area, and in particular that this is invariant under the flow of NLS.

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