Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

✍ Scribed by Michael Voit

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 679 KB
• Volume: 59
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1996.0063

✦ Synopsis

For n 2, let (+ x {, n ) { 0 be the distributions of the Brownian motion on the unit sphere S n /R n+1 starting in some point x # S n . This paper supplements results of Saloff-Coste concerning the rate of convergence of + x {, n to the uniform distribution U n on S n for { Ä depending on the dimension n. We show that, lim n Ä &+ x { n, n &U n &= 2 } erf(e &s Â-8) for { n :=(ln n+2s)Â(2n), where erf denotes the error function. Our proof depends on approximations of the measures + x {, n by measures which are known explicitly via Poisson kernels on S n , and which tend, after suitable projections and dilatations, to normal distributions on R for n Ä . 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.
1996 Academic Press, Inc.

1. ASYMPTOTIC BEHAVIOR OF GAUSSIAN MEASURES ON n-SPHERES

For n 2 let U n be the uniform distribution on the n-sphere S n /R n+1 . If L n is the Laplace Beltrami operator on S n , then (H {, n ) { 0 :=(e &{Ln ) { 0 forms a Markovian selfadjoint semigroup of operators on L P (S n , U n ) (1 p ) which may be regarded as semigroup of operators related to Brownian motion on S n . The semigroup (H {, n ) { 0 admits a kernel (H {, n ) { 0 with 1) and with 00, x, y # S n . In particular, for each x # S n , the functions h x {, n ( y) :=h {, n (x, y) are the L 1 -densities of the semigroup (+ x {, n :=h x {, n U n ) { 0 of the distributions of the Brownian motion on S n starting in x at time 0.
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