Anderson parabolic model for a quasi-stationary medium

We study the Anderson Parabolic Model for a random medium which is a product of an i.i.d. space-like random field and a white noise. The model has long range space-time correlations and is intermediate between the stationary case and the "turbulent" one, which were studied in previous works. Under some natural assumptions on the distribution of the space potential, we prove existence and uniqueness, and derive the long time asymptotics for the annealed moments, and the "semi-annealed" ones, for which expectation is taken only w.r.t. the white noise. A conjecture for the fully quenched case is discussed on a simplified model.