Convolution Semigroups of Probability Measures on Gelfand Pairs, Revisited

TL;DR

This paper explores convolution semigroups on Lie groups related to symmetric spaces, clarifies invariance properties, and introduces a new class derived from Cartan decomposition suitable for modeling geodesic motion.

Contribution

It identifies a novel class of convolution semigroups using Cartan decomposition and stochastic differential equations, expanding understanding of processes on symmetric spaces.

Findings

We show that weaker right K-invariance coincides with the usual notion.

Generalized negative definite functions do not yield new semigroups.

A new class of semigroups models geodesic motion in symmetric spaces.

Abstract

Our goal is to find classes of convolution semigroups on Lie groups $G$ that give rise to interesting processes in symmetric spaces $G/K$. The $K$-bi-invariant convolution semigroups are a well-studied example. An appealing direction for the next step is to generalise to right $K$-invariant convolution semigroups, but recent work of Liao has shown that these are in one-to-one correspondence with $K$-bi-invariant convolution semigroups. We investigate a weaker notion of right $K$-invariance, but show that this is, in fact, the same as the usual notion. Another possible approach is to use generalised notions of negative definite functions, but this also leads to nothing new. We finally find an interesting class of convolution semigroups that are obtained by making use of the Cartan decomposition of a semisimple Lie group, and the solution of certain stochastic differential equations. Examples suggest that these are well-suited for generating random motion along geodesics in symmetric spaces.