2010/06/30 by David Applebaum
Mathematics · #Advanced Topics in Algebra #Convolution (computer science) #Distribution (mathematics) #Heat kernel #Kernel (algebra) #Laplace transform #Locally compact space #Mathematical Analysis and Transform Methods #Measure (data warehouse) #Random Matrices and Applications #Real line #Semigroup #TRACE (psycholinguistics) #math.FA #math.GR #math.PR
paper · pdf · doi:10.1214/10-aop604
published as Annals of Probability 2011, Vol. 39, No. 6, 2474-2496 · Published in at http://dx.doi.org/10.1214/10-AOP604 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2011/11/01 · arxiv created 2012/02/13 · arxiv updated 2012/02/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.