随机介质中扩散过程的尺度跃迁

Scaling-up of DiffusionProcess in a Random Medium

本文考虑随机多孔介质中的示踪粒子的随机移动过程和相应的尺度跃迁问题 .假设当时间和空间进行适当的尺度跃迁时 ,其粒子的移运过程弱收敛于是 d-维中心布朗运动 ,具有协方差 D.随机介质对示踪粒子的作用可表示为小的扰动力 ,扰动过程收敛于具有相同协方差阵的布郎运动 ,但具有一个形如 M.a的附加漂移 .对于扰动的粒子的稀薄过程 ,我们证明了试验粒子的流度和协方差通过 Einstein公式相关联 .证明 Einstein公式所用的方法就是计算轨迹空间上的测度的 Radon-Nikodym导数 (Girsanov公式 ) .研究单个粒子在具有时间独立的随机非均匀性质的格上运动和在速度满足 Langevin方程的随机势场中的运动 ,关于尺度跃迁过程得到了一些特征性质和扩散矩阵和漂移之间的关系 .

Consider the stochastic displacement process of a tagged partical X(t) in random porous medium and scaling-up problem. Assume that this displacement process converges weakly to d-dimensional centered Brownnian motion with covariance D, when space and time are appropriately sacled. The influence of random medium can be cast into a small pertubation force, the pertubed process converges to Brownian motion having the same covariance D and an additional drift of the from M.a. We show that the mobility of test particle and covariance are related to each other by the Einstein formula. The method used to verlfy Einstain formula is the calculus of Radon-Nikodym derivatives of measures in the space of trajectories (Cirsanov′s formula). We study the motion of a single particle on a lattice with time independent random inhomogeneitiees and and in a random potentiail whose velocity is governed by a Langevin equation, obtain some properties of the scaling-up process and relation between diffusion matriix and drift.