Homogenization of Heat Equation with Large Time-dependent Random Potential

Stochastic Processes and their Applications (Impact Factor: 1.06). 01/2014; 125(1). DOI: 10.1016/
Source: arXiv


This paper concerns the homogenization problem of heat 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.

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    ABSTRACT: This paper reviews several results obtained recently in the convergence of solutions to elliptic or parabolic equations with large highly oscillatory random potentials. Depending on the correlation properties of the potential, the resulting limit may be either deterministic and solution of a homogenized equation or random and solution of a stochastic PDE. In the former case, the residual random fluctuations of the heterogeneous solution may also be characterized, or at least the rate of convergence to the deterministic limit established. We present several results that can be obtained by the methods of asymptotic perturbations, diagrammatic expansions, probabilistic representations, and the multiscale method.
    Communications in mathematical sciences 01/2015; 13(3):729-748. DOI:10.4310/CMS.2015.v13.n3.a7 · 1.12 Impact Factor