A decomposition of Markov processes via group actions

TL;DR

This paper introduces a novel decomposition of Markov processes on manifolds with Lie group symmetries into radial and angular components, extending classical Euclidean results to more general geometric settings.

Contribution

It provides a new framework for decomposing Markov processes on manifolds with group actions, including a representation of conditioned angular parts as nonhomogeneous Lévy processes.

Findings

Angular part conditioned on radial path is a nonhomogeneous Lévy process

Extension of Euclidean Brownian motion skew-product to manifolds

Decomposition facilitates analysis of Markov processes on symmetric spaces

Abstract

We study a decomposition of a general Markov process in a manifold invariant under a Lie group action into a radial part (transversal to orbits) and an angular part (along an orbit). We show that given a radial path, the conditioned angular part is a nonhomogeneous \levy process in a homogeneous space, we obtain a representation of such processes, and as a consequence, we extend the well known skew-product of Euclidean Brownian motion to a general setting.