有界面的周期均匀化：多维情况
Periodic homogenization with an interface: The multi-dimensional case
摘要：
我们考虑一个扩散过程的系数是周期性的“界面区域”的有限厚度外。在这篇文章中调查的问题是限制长时间/大尺度的行为，这样一个过程下扩散重新标度。很明显，在界面之外，极限过程必须表现得像布朗运动，扩散矩阵由标准均匀化理论给出。因此，有趣的行为发生在接口上。我们的主要结果是，极限过程是一个半鞅的有界变化部分是成比例的本地时间花在界面上。比例矢量可以具有平行于界面的非零分量，使得限制扩散不一定是可逆的。我们还展示了一个明确的方式来确定其参数的原始扩散的系数。类似于一维的情况下，我们的证明方法依赖于框架提供的Freidlin和Wentzell [安。21（1993）2215--2245]，以便识别限制过程的生成器。
英文摘要：
We consider a diffusion process with coefficients that are periodic outside of an "interface region" of finite thickness. The question investigated in this article is the limiting long time/large scale behavior of such a process under diffusive rescaling. It is clear that outside of the interface, the limiting process must behave like Brownian motion, with diffusion matrices given by the standard theory of homogenization. The interesting behavior therefore occurs on the interface. Our main result is that the limiting process is a semimartingale whose bounded variation part is proportional to the local time spent on the interface. The proportionality vector can have nonzero components parallel to the interface, so that the limiting diffusion is not necessarily reversible. We also exhibit an explicit way of identifying its parameters in terms of the coefficients of the original diffusion. Similarly to the one-dimensional case, our method of proof relies on the framework provided by Freidlin and Wentzell [Ann. Probab. 21 (1993) 2215--2245] for diffusion processes on a graph in order to identify the generator of the limiting process.
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