Homogenization of Lateral Diffusion on a Random Surface

We study the problem of lateral diffusion on a static, quasi-planar surface generated by a stationary, ergodic random field possessing rapid small-scale spatial fluctuations. The aim is to study the effective behavior of a particle undergoing Brownian motion on the surface, viewed as a projection on the underlying plane. By formulating the problem as a diffusion in a random medium, we are able to use known results from the theory of stochastic homogenization of SDEs to show that, in the limit of small scale fluctuations, the diffusion process behaves quantitatively like a Brownian motion with constant diffusion tensor $\mathbf{D}$. In one dimension, the effective diffusion coefficient is given by $\frac{1}{Z^2}$, where $Z$ is the average line element of the surface. In two-dimensions, $\mathbf{D}$ will not have a closed-form expression in general. However, we are able to derive variational bounds for the effective diffusion tensor. Moreover, in the special case when $\mathbf{D}$ is isotropic, we show that $\mathbf{D}=\frac{1}{Z}\mathbf{I}$, where $Z$ is the average area element of the random surface. We also describe a numerical scheme for approximating the effective diffusion tensor and illustrate this scheme with three examples.