Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn[greater-or-equal, slanted]2, let ([mu]x[tau], n)[tau][greater-or-equal, slanted]0be the distributions of the Brownian motion on the unit sphereSn[subset of]n+1starting in some pointx[set membership, variant]Sn. This paper supplements results of Saloff-Coste concerning the rate of convergence of[mu]x[tau], nto the uniform distributionUnonSnfor[tau]-->[infinity] depending on the dimensionn. We show that,[formula]for[tau]n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measures[mu]x[tau], nby measures which are known explicitly via Poisson kernels onSn, and which tend, after suitable projections and dilatations, to normal distributions on forn-->[infinity]. 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.