Brownian motion on a sphere: distribution of solid angles

We study the diffusion of Brownian particles on the surface of a sphere and compute the distribution of solid angles enclosed by the diffusing particl es. This function describes the distribution of geometric phases in two-sta te quantum systems (or polarized light) undergoing random evolution. Our re sults are also relevant to recent experiments which observe the Brownian mo tion of molecules on curved surfaces such as micelles and biological membra nes. Our theoretical analysis agrees well with the results of computer expe riments.