Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn⩾2, let (μxτ, n)τ⩾0be the distributions of the Brownian motion on the unit sphereSn⊂Rn+1starting in some pointx∈Sn. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμxτ, nto the uniform distributionUnonSnforτ→∞ depending on the dimensionn. We show that,[formula]forτn:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμxτ, nby measures which are known explicitly via Poisson kernels onSn, and which tend, after suitable projections and dilatations, to normal distributions on R forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.