Random Walks on Hyperspheres of Arbitrary Dimensions

We consider random walks on the surface of the sphere $S_{n-1}$ ($n\ge 2$) 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 $validin$E_{n-1}$shouldbereplacedin$S_{n-1}$ by the generic law $<\cos \theta >\propto \exp (-t\mathrm{/}\tau )$, where $\theta$ denotes the angular displacement of the walker. More generally one has $<C_L^{n\mathrm{/}2-1}\cos (\theta )>\propto \exp (-t\mathrm{/}\tau (L,n))$ where $C_L^{n\mathrm{/}2-1}$ a Gegenbauer polynomial. Conjectures concerning random walks on a fractal inscribed in $S_{n-1}$ are given tentatively.