Convolution of probability measures on Lie groups and homogenous spaces

TL;DR

This paper investigates continuous convolution semigroups of probability measures on Lie groups and homogeneous spaces, extending existing results to include these spaces and characterizing their structure.

Contribution

It extends the theory of convolution semigroups to homogeneous spaces and provides a decomposition result for such semigroups on Lie groups and spaces.

Findings

Convolution semigroups can be decomposed into initial measure and a continuous semigroup.

Extension of Dani-McCrudden's embedding result to homogeneous spaces.

Characterization of convolution semigroups on Lie groups and homogeneous spaces.

Abstract

We study (weakly) continuous convolution semigroups of probability measures on a Lie group G or a homogeneous space G/K, where K is a compact subgroup. We show that such a convolution semigroup is the convolution product of its initial measure and a continuous convolution semigroup with initial measure at the identity of G or the origin of G/K. We will also obtain an extension of Dani-McCrudden's result on embedding an infinitely divisible probability measure in a continuous convolution semigroup on a Lie group to a homogeneous space.