On invariant Gibbs measures conditioned on mass and momentum

Abstract

We construct a Gibbs measure for the nonlinear Schrodinger equation (NLS) on the circle, conditioned on prescribed mass and momentum: d μ_{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 μ_{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.

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