Homogenization with Large Spatial Random Potential

Guillaume Bal
2010 Multiscale Modeling & simulation  
We consider the homogenization of parabolic equations with large spatiallydependent 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 ≥ m, where m is the order
more » ... here m is the order of the spatial pseudo-differential operator in the parabolic equation. In dimension d < m, the solution to the parabolic equation is shown to converge to the (non-deterministic) solution of a stochastic equation in [2] . The results are then extended to cover the case of long range random potentials, which generate larger, but still asymptotically Gaussian, random fluctuations. keywords: Homogenization theory, partial differential equations with random coefficients, Gaussian fluctuations, large potential, long range correlations AMS: 35R60, 35P20, 60H05, 35K15.
doi:10.1137/090754066 fatcat:dp5m6roskfhszfvu2y5bhqfese