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.
"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