When do skew-products exist?

TL;DR

This paper examines the conditions under which skew-product decompositions of diffusions on manifolds exist, especially focusing on one-dimensional homogeneous spaces where classical assumptions may fail or need refinement.

Contribution

It presents counterexamples to a known theorem on skew-product decompositions, highlighting the nuanced conditions needed for such decompositions in one-dimensional cases.

Findings

Counterexamples show the failure of classical skew-product decomposition in certain cases

The validity of the theorem depends on the dimension of the homogeneous space

Conditions for the existence of a standard skew-product decomposition are clarified

Abstract

The classical skew-product decomposition of planar Brownian motion represents the process in polar coordinates as an autonomously Markovian radial part and an angular part that is an independent Brownian motion on the unit circle time-changed according to the radial part. Theorem 4 of Liao (2009) gives a broad generalization of this fact to a setting where there is a diffusion on a manifold $X$ with a distribution that is equivariant under the smooth action of a Lie group $K$. Under appropriate conditions, there is a decomposition into an autonomously Markovian "radial" part that lives on the space of orbits of $K$ and an "angular" part that is an independent Brownian motion on the homogeneous space $K/M$, where $M$ is the isotropy subgroup of a point of $x$, that is time-changed with a time-change that is adapted to the filtration of the radial part. We present two apparent counterexamples to Theorem 4 of Liao (2009). In the first counterexample the angular part is not a time-change of any Brownian motion on $K/M$, whereas in the second counterexample the angular part is the time-change of a Brownian motion on $K/M$ but this Brownian motion is not independent of the radial part. In both of these examples $K/M$ has dimension $1$. The statement and proof of Theorem 4 from Liao (2009) remain valid when $K/M$ has dimension greater than $1$. Our examples raise the question of what conditions lead to the usual sort of skew-product decomposition when $K/M$ has dimension $1$ and what conditions lead to there being no decomposition at all or one in which the angular part is a time-changed Brownian motion but this Brownian motion is not independent of the radial part.