Almost Sure Scattering for the One Dimensional Nonlinear Schrodinger Equation

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p > 1. On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation, and it is sometimes possible to modify them suitably and get invariant or quasi-invariant measures for the non linear problem.
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