Random walks on hyperspheres of arbitrary dimensions

We consider random walks on the surface of the sphere Sn-1 (n greater than or equal to 2) of the n-dimensional Euclidean space E-n, in short a hypersphere. By solving the diffusion equation-in Sn-1 we show that the usual law <r(2)> proportional to t valid in En-1 should be replaced in Sn-1 by the generic law <costheta> proportional to exp(-t/tau), where theta denotes the angular displacement of the walker. More generally one has <C-L(n/2-1)(cos theta)> proportional to exp(-t/tau(L,n)) where C-L(n/2-1) is a Gegenbauer polynomial. Conjectures concerning random walks on a fractal inscribed in Sn-1 are given tentatively.