On the embeddability of certain infinitely divisible probability measures on Lie groups

TL;DR

This paper establishes conditions under which infinitely divisible probability measures on certain Lie groups can be embedded into continuous convolution semigroups, advancing understanding of their structure and embeddability.

Contribution

It provides new sufficient conditions for embeddability of infinitely divisible measures on Lie groups, including specific classes like the Walnut group.

Findings

All infinitely divisible measures on the Walnut group are embeddable.

Embeddability is confirmed under various new conditions for certain Lie groups.

The paper extends previous results by identifying broader classes of embeddable measures.

Abstract

We describe certain sufficient conditions for an infinitely divisible probability measure on a class of connected Lie groups to be embeddable in a continuous one-parameter convolution semigroup of probability measures. (Theorem 1.3). This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain Lie groups, including the so called Walnut group (Corollary 1.5). The embeddability is concluded also under certain other conditions (Corollary 1.4 and Theorem 1.6).