On invariant Gibbs measures conditioned on mass and momentum
Probability
2011-07-25 v2 Analysis of PDEs
Abstract
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.
Keywords
Cite
@article{arxiv.1012.3432,
title = {On invariant Gibbs measures conditioned on mass and momentum},
author = {Tadahiro Oh and Jeremy Quastel},
journal= {arXiv preprint arXiv:1012.3432},
year = {2011}
}
Comments
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. Math. Soc. Japan