  This paper concerns the homogenization problem of a parabolic equation with
large, time-dependent, random potentials in high dimensions $d\geq 3$.
Depending on the competition between temporal and spatial mixing of the
randomness, the homogenization procedure turns to be different. We characterize
the difference by proving the corresponding weak convergence of Brownian motion
in random scenery. When the potential depends on the spatial variable
macroscopically, we prove a convergence to SPDE.
