Random walk on a sphere and on a Riemannian manifold

Abstract

A random walk on a sphere consists of a chain of random steps for which all directions from the starting point are equally probable, while the length a of the step is either fixed or subject to a given probability distribution p(a). The discussion allows the fixed length a or given distribution p(<x), to vary from one step of the chain to another. A simple formal solution is obtained for the distribution of the moving point after any random walk ; the simplicity depends on the fact that the individual steps commute and therefore have common eigenfunctions. Results are derived on the convergence of the eigenfunction expansion and on the asymptotic behaviour after a large number of random steps. The limiting case of diffusion is discussed in some detail and compared with the distribution propounded by Fisher (1953). The corresponding problem of random walk on a general Riemannian manifold is also attacked. It is shown that commutability does not hold in general, but that it does hold in completely harmonic spaces and in some others. In commutative spaces, complete analogy with the method employed for a sphere is found.