大空间随机势均匀化
Homogenization with large spatial random potential
摘要：
本文研究了具有高斯随机场模型的大空间势的抛物方程的均匀化问题。在随机势的相关长度为零的极限下，导出了均匀化方程。我们刻画了随机涨落中的主导效应，并证明了它们的空间矩收敛于高斯随机变量。这两个结果都适用于足够小的时间和足够大的空间维度$d\geq\m$，其中$\m$是抛物方程中空间伪微分算子的阶。在$d<\m$维中，抛物方程的解收敛到随机方程的（非确定性）解.结果，然后扩展到覆盖的情况下，产生较大的，但仍然渐近高斯，随机波动的长程随机电位。
英文摘要：
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 $d\geq\m$, where $\m$ 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 the companion paper [2]. The results are then extended to cover the case of long range random potentials, which generate larger, but still asymptotically Gaussian, random fluctuations.
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