Infinitely divisible central probability measures on compact Lie groups---regularity, semigroups and transition kernels

Details Infinitely divisible central probability measures on compact Lie groups---regularity, semigroups and transition kernels Infinitely divisible central probability measures on compact Lie groups---regularity, semigroups and transition kernels

2010-06-24

arxiv.org/abs/1006.4711v3

Annals of Probability 2011, Vol. 39, No. 6, 2474-2496

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP604 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP604

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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