  Cubical complexes are metric spaces constructed by gluing together unit cubes
in an analogous way to the construction of simplicial complexes. We construct
Brownian motion on such spaces, define random walks, and prove that the
transition kernels of the random walks converge to that for Brownian motion.
The proof involves pulling back onto the complex the distribution of Brownian
sample paths on the standard cube, and combining this with a distribution on
walks between cubes in the complex. The main application lies in analysing sets
of evolutionary trees: several tree spaces are cubical complexes and we briefly
describe our results and some applications in this context. Our results extend
readily to a class of polyhedral complex in which every cell of maximal
dimension is isometric to a given fixed polyhedron.
