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

TL;DR

This paper introduces a class of infinitely divisible probability measures on compact Lie groups, explores their regularity and semigroup properties, and analyzes the short-time behavior of their densities, revealing new insights beyond classical heat kernels.

Contribution

It defines a new class of symmetric infinitely divisible measures on compact Lie groups via the Casimir operator and characterizes their regularity, semigroup generators, and transition densities.

Findings

Conditions for smooth densities of the measures

Representation of semigroups as pseudo-differential operators

Distinct short-time asymptotics for Cauchy distributions on specific groups

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.