Random Walks on Hyperspheres of Arbitrary Dimensions

We consider random walks on the surface of the sphere $S_{n-1}$ ($n \geq 2$) of the $n$-dimensional Euclidean space $E_n$, in short a hypersphere. By solving the diffusion equation in $S_{n-1}$ we show that the usual law $<r^2 > \varpropto t$ valid in $E_{n-1}$ should be replaced in $S_{n-1}$ by the generic law $<\cos \theta > \varpropto \exp(-t/\tau)$, where $\theta$ denotes the angular displacement of the walker. More generally one has $<C^{n/2-1}_{L}\cos(\theta)> \varpropto \exp(-t/ \tau(L,n))$ where $C^{n/2-1}_{L}$ a Gegenbauer polynomial. Conjectures concerning random walks on a fractal inscribed in $S_{n-1}$ are given tentatively.