  This paper concerns the macroscopic behavior of solutions to parabolic
equations with large, highly oscillatory, random potential. When the
correlation function of the random potential satisfies a specific integrability
condition, we show that the random solution converges, as the correlation
length of the medium tends to zero, to the deterministic solution of a
homogenized equation in dimension $d\geq3$. Our derivation is based on a
Feynman-Kac probabilistic representation and the Kipnis-Varadhan method applied
to weak convergence of Brownian motions in random sceneries. For sufficiently
mixing coefficients, we also provide an optimal rate of convergence to the
homogenized limit using a quantitative martingale central limit theorem. As
soon as the above integrability condition fails, the solution is expected to
remain stochastic in the limit of a vanishing correlation length. For a large
class of potentials given as functionals of Gaussian fields, we show the
convergence of solutions to stochastic partial differential equations (SPDE)
with multiplicative noise. The Feynman-Kac representation and the corresponding
weak convergence of Brownian motions in random sceneries allows us to explain
the transition from deterministic to stochastic limits as a function of the
correlation function of the random potential.
