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Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions
Authors:Michael Voit
Affiliation:Mathematisches Institut, Universität Tübingen, Tübingen, Germany
Abstract:Forn2, let (μxτn)τ0be the distributions of the Brownian motion on the unit sphereSnn+1starting in some pointxSn. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμxτnto 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 measuresμxτnby measures which are known explicitly via Poisson kernels onSn, and which tend, after suitable projections and dilatations, to normal distributions on 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.
Keywords:Gaussian measures   ultraspherical polynomials   hypergroups   convergence to equilibrium   total variation distance   central limit theorem
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