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November 2011 Infinitely divisible central probability measures on compact Lie groups—regularity, semigroups and transition kernels
David Applebaum
Ann. Probab. 39(6): 2474-2496 (November 2011). DOI: 10.1214/10-AOP604

Abstract

We introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator. The class includes Gauss, Laplace and stable-type measures. We find conditions for such a measure to have a smooth density and give examples. The Hunt semigroup and generator of convolution semigroups of measures are represented as pseudo-differential operators. For sufficiently regular convolution semigroups, the transition kernel has a tractable Fourier expansion and the density at the neutral element may be expressed as the trace of the Hunt semigroup. We compute the short time asymptotics of the density at the neutral element for the Cauchy distribution on the d-torus, on SU(2) and on SO(3), where we find markedly different behaviour than is the case for the usual heat kernel.

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David Applebaum. "Infinitely divisible central probability measures on compact Lie groups—regularity, semigroups and transition kernels." Ann. Probab. 39 (6) 2474 - 2496, November 2011. https://doi.org/10.1214/10-AOP604

Information

Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1237.60005
MathSciNet: MR2932674
Digital Object Identifier: 10.1214/10-AOP604

Subjects:
Primary: 35K08 , 43A05 , 47D07 , 60B15 , 60G51

Keywords: Casimir operator , central measure , compact Lie group , convolution semigroup , Hunt semigroup , Infinite divisibility , ‎pseudo-differential operator , Sobolev space , symbol , Transition density

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 6 • November 2011
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