On stationary actions of locally compact groups and their Radon-Nikodym cocycles

TL;DR

This paper investigates stationary actions of locally compact groups, revealing their dynamical properties, regularity of Radon-Nikodym cocycles, and constructing models that incorporate these cocycles, with implications for harmonic analysis.

Contribution

It extends Furstenberg-Glasner's theorem to noncompact groups, introduces harmonic majorants for Radon-Nikodym derivatives, and constructs a universal Radon-Nikodym model strengthening Mackey-Varadarajan's theorem.

Findings

Every stationary action is conservative.

Stationary actions with an absolutely continuous invariant measure are probability preserving.

Constructed a Poisson boundary with unbounded Poisson kernel, failing Kaimanovich's SAT* property.

Abstract

We study stationary actions of locally compact measured groups through the structure and regularity of their Radon-Nikodym cocycles. We start with two dynamical consequences of stationarity. Extending a theorem of Furstenberg-Glasner from discrete groups to noncompact locally compact groups, we show that every stationary action is conservative. Thus stationary actions are never of type I. We then show that an ergodic stationary action admitting an absolutely continuous invariant sigma-finite measure is in fact probability preserving. Thus stationary actions are never of type II_infty. Using a construction of Katznelson-Weiss and Vaes-Verjans, we show that if a group admits a stationary action of type III_1, then it admits stationary actions of every type III_lambda. The second part concerns the regularity of the Radon-Nikodym cocycle. We introduce the harmonic majorant on normalized positive harmonic functions, which gives Harnack-type control of Radon-Nikodym derivatives of stationary actions. For compactly supported probability measures with an Lp-density for some p>1, we prove that the harmonic majorant is finite and locally bounded. As a consequence, such measured groups admit a universal compact Radon-Nikodym model: a single compact G-space with a continuous cocycle into which stationary actions can be realized, so that the ambient cocycle gives a version of its Radon-Nikodym cocycle. This strengthens the Mackey-Varadarajan compact model theorem by incorporating the Radon-Nikodym cocycle into the model. By contrast, we construct a random walk on the real affine group whose Poisson boundary fails Kaimanovich's SAT* property: the Poisson kernel is unbounded arbitrarily close to the identity. Therefore, Harnack's inequality already fails for positive harmonic functions. In particular, this Poisson boundary admits no topological model with continuous Poisson kernel.