Homogenization of random parabolic operator with large potential

Abstract

We study the averaging problem for a divergence form random parabolic operators with a large potential and with coefficients rapidly oscillating both in space and time variables. We assume that the medium possesses the periodic microscopic structure while the dynamics of the system is random and, moreover, diffusive. A parameter α will represent the ratio between space and time microscopic length scales. A parameter β will represent the effect of the potential term. The relation between α and β is of great importance. In a trivial case the presence of the potential term will be “neglectable”. If not, the problem will have a meaning if a balance between these two parameters is achieved, then the averaging results hold while the structure of the limit problem depends crucially on α (with three limit cases: one classical and two given under martingale problems form). These results show that the presence of stochastic dynamics might change essentially the limit behavior of solutions.