  The macroscopic transport properties in a disordered potential, namely
diffusion and weak/strong localization, closely depend on the microscopic and
statistical properties of the disorder itself. This dependence is rich of
counter-intuitive consequences. It can be particularly exploited in matter wave
experiments, where the disordered potential can be tailored and controlled, and
anisotropies are naturally present. In this work, we apply a perturbative
microscopic transport theory and the self-consistent theory of Anderson
localization to study the transport properties of ultracold atoms in
anisotropic 2D and 3D speckle potentials. In particular, we discuss the
anisotropy of single-scattering, diffusion and localization. We also calculate
a disorder-induced shift of the energy states and propose a method to include
it, which amounts to renormalize energies in the standard on-shell
approximation. We show that the renormalization of energies strongly affects
the prediction for the 3D localization threshold (mobility edge). We illustrate
the theoretical findings with examples which are revelant for current matter
wave experiments, where the disorder is created with a laser speckle. This
paper provides a guideline for future experiments aiming at the precise
location of the 3D mobility edge and study of anisotropic diffusion and
localization effects in 2D and 3D.
