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: dµa,b = Z −1 1{ ´ T |u|2=a}1{iT uux=b} e ± 1 ´ T |u| p− 1 ´ T |u| 2 dP for a 2 R + and b 2 R, where P is the complex-valued Wiener measure on the circle. We also show that µa,b is invariant under the flow of NLS. We note that iT uux is the Levy stochastic area, and in particular that this is invariant under the flow of NLS.