  We examine theoretically the effects of random topographical substrates on
the motion of two-dimensional droplets via appropriate statistical approaches.
Different random substrate families are represented as stationary random
functions. The variance of the droplet shift at both early times and in the
long-time limit is deduced and the droplet footprint is found to be a normal
random variable at all times. It is shown that substrate roughness decreases
droplet wetting, illustrating also the tendency of the droplet to slide without
spreading as equilibrium is approached. Our theoretical predictions are
verified by numerical experiments.
