Random integrals and correctors in homogenization
Random integrals and correctors in homogenization
复制标题
均质化中的随机积分和校正器
作者：
G. Bal;J. Garnier;Sébastien Motsch;V. Perrier
本文讨论一维随机系数振荡的椭圆型方程的齐次化问题。众所周知，在随机介质中，椭圆方程的随机解在相关长度为零的极限下收敛于有效介质椭圆型方程的解。众所周知，当随机介质中的相关性足够短时，均匀化的校正器，即随机解和均匀化解之间的差，在分布上收敛于高斯过程。此外，极限过程可以写成关于标准布朗运动的随机积分。我们将这一结果推广到一大类具有长程关联的过程。在这种情况下，校正器也收敛到高斯随机过程，其具有关于分数布朗运动的随机积分的解释。此外，我们还证明了关联范围越大，校正器的幅度越大。导数是基于对具有长程相关性的过程的随机振荡积分的仔细分析。我们还利用了一维椭圆型方程的解的显式表达式。
This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well-known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well-known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.