Anderson Localization for Time Quasi Periodic Random Sch\"odinger and Wave Operators

We prove that at large disorder, with large probability and for a set of Diophantine frequencies of large measure, Anderson localization in ${\mathbb{Z}}^d$ is {\it stable} under localized time-quasi-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The main tools are the Fr"ohlich-Spencer mechanism for the random component and the Bourgain-Goldstein-Schlag mechanism for the quasi-periodic component. The formulation of this problem is motivated by questions of Anderson localization for non-linear Schr"odinger equations.