  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 behaviour 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 $D$. While $D$ will not have a closed-form expression in general, we are
able to derive variational bounds for the effective diffusion tensor, and using
a duality transformation argument, obtain a closed form expression for $D$ in
the special case where $D$ is isotropic. We also describe a numerical scheme
for approximating the effective diffusion tensor and illustrate this scheme
with two examples.
