Long Time Anderson Localization for the Nonlinear Random Schrodinger Equation

We prove long time Anderson localization for the nonlinear random Schrodinger equation for arbitrary l(2) initial data, hence giving an answer to a widely debated question in the physics community. The proof uses a Birkhoff normal form type transform to create a barrier where there is essentially no propagation. One of the new features is that this transform is in a small neighborhood enabling us to treat "rough" data, where there are no moment conditions. The formulation of the present result is inspired by the RAGE theorem.