Homogenization with Large Spatial Random Potential

Abstract

We consider the homogenization of parabolic equations with large spatially dependent potentials modeled as Gaussian random fields. We derive the homogenized equations in the limit of vanishing correlation length of the random potential. We characterize the leading effect in the random fluctuations and show that their spatial moments converge in law to Gaussian random variables. Both results hold for sufficiently small times and in sufficiently large spatial dimensions $d\geq\mathfrak{m}$, where $\mathfrak{m}$ is the order of the spatial pseudodifferential operator in the parabolic equation. In dimension $d<\mathfrak{m}$, the solution to the parabolic equation is shown to converge to the (nondeterministic) solution of a stochastic equation in [G. Bal, Comm. Math. Phys., 212 (2009), pp. 457–477]. The results are then extended to cover the case of long range random potentials, which generate larger, but still asymptotically Gaussian, random fluctuations.