Anderson localization for time quasi-periodic random Schrodinger and wave equations

We prove that at large disorder, with large probability and for a corresponding set of Diophantine frequencies of large measure, Anderson localization in Z(d) is stable under localized time quasi-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The main tools are the Frohlich-Spencer mechanism for the random component and the Bourgain-Goldstein-Schlag mechanism for the quasi-periodic component. This paper paves the way for the construction of time quasi-periodic KAM type of solutions of non linear random Schrodinger equations in [BW].