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

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.

• The paper establishes a full classification of dynamical types for ergodic stationary actions, proving that all stationary actions of noncompact lcsc groups are conservative and probability-preserving under specific conditions.
• It introduces a harmonic majorant that uniformly controls Radon-Nikodym cocycles by establishing a Harnack-type inequality under integrability and compact support assumptions.
• The study constructs a universal compact model embedding continuous cocycles, thereby extending classical results from the discrete to the locally compact setting.

Stationary Actions of Locally Compact Groups and Radon-Nikodym Cocycles

Overview

The paper "On stationary actions of locally compact groups and their Radon-Nikodym cocycles" (2604.09093) investigates the structure, regularity, classification, and topological realization of stationary actions of locally compact second countable (lcsc) groups equipped with admissible probability measures. The core focus lies on the behavior and analytic properties of Radon-Nikodym cocycles associated with these actions, the possible dynamical types of stationary systems, and the construction of universal compact models encapsulating the cocycle data. The results extend and deepen several classical theorems from the discrete group context to the more analytically and topologically complicated setting of lcsc groups.

Structural Properties and Dynamical Type Classification

A principal achievement is the full classification of possible dynamical types for ergodic stationary actions of noncompact measured groups. By extending the Furstenberg–Glasner theorem, the authors prove that all stationary actions of noncompact lcsc groups are conservative (Theorem A). This eliminates type I (completely dissipative) possibility for such actions. A key rigidity result is established: any ergodic stationary action that has an absolutely continuous invariant σ-finite measure is necessarily probability-preserving (Theorem B). Consequently, ergodic stationary actions cannot be of type II$_\infty$.

Moreover, utilizing constructions by Katznelson–Weiss and Vaes–Verjans, the authors show that, whenever a stationary action of type III$_1$ exists for a given group, then stationary actions of every type III$_\lambda$ ($0 \leq \lambda \leq 1$) can also be constructed (extension and realization of Krieger's types for stationary actions). This generalizes the classical case for $\mathbb{Z}$-actions to all lcsc groups and demonstrates the full spectrum of type III actions is attainable.

Radon-Nikodym Cocycle Regularity and Harmonic Majorant

A central analytic object in the theory of stationary actions is the Radon-Nikodym cocycle, whose behavior encodes nonsingularity and classification properties. This paper introduces the harmonic majorant as a universal control function on $\theta$-normalized positive harmonic functions: for $g \in G$, define

$$\mathfrak{m}_\theta(g) = \sup\{u(g) : u \in \mathbb{H}(G,\theta)\}$$

where $\mathbb{H}(G,\theta)$ is the normalized cone of positive $\theta$-harmonic functions $_1$0 with $_1$1 and $_1$2. This majorant quantifies the pointwise maximal distortion possible under stationary cocycles.

A significant result is a Harnack-type inequality: If $_1$3 admits a density with compact support in $_1$4 for some $_1$5, then $_1$6 is finite and locally bounded everywhere (Theorem C), implying uniform control over all possible Radon-Nikodym derivatives in all stationary actions. The converse is shown sharp: In the real affine group, the authors construct a measure $_1$7 whose Poisson boundary fails Kaimanovich's $_1$8 property; Radon-Nikodym derivatives become unbounded arbitrarily close to the identity. This demonstrates the necessity of integrability and support conditions for any universal regularity statement.

Compact Universal Models for Cocycles

Moving to topological realization, the paper establishes that when the harmonic majorant is $_1$9-integrable, there exists a universal compact $_\lambda$0-space and a continuous cocycle into which every stationary action (with its cocycle) can be embedded. This universal compact model refines and strengthens the Mackey–Varadarajan theory by ensuring that the Radon-Nikodym cocycle is continuous in the model, and every stationary system is realized inside a fixed compact system with continuous cocycle. The construction uses the compact-open topology on the space of normalized $_\lambda$1-harmonic functions and produces an explicit universal object.

The authors show that such a universal model always exists for measured groups where $_\lambda$2 has compactly supported $_\lambda$3-density for $_\lambda$4 (Corollary D). In compactly generated groups, explicit growth and integrability conditions on the harmonic majorant yield practical criteria for compact model existence.

Failure of Universal Regularity: Non-$_\lambda$5 Examples

In a detailed example, the paper proves that for certain admissible measures on the real affine group, the Poisson boundary fails even the local boundedness of the Radon-Nikodym kernel. This is accomplished through precise probabilistic computations of random walks with carefully selected increment distributions, demonstrating that topological or continuous models for the cocycle cannot always exist—highlighting a strict dichotomy in the locally compact case versus the countable discrete case.

Implications and Prospects

The paper’s results have multiple theoretical and practical implications:

• Rigidity and Classification: The sharp type rigidity for stationary actions impacts the possible classification strategies for group actions and highlights profound differences between lcsc groups and discrete groups.
• Harmonic Analysis and Random Walks: The harmonic majorant and Harnack-type statements bridge harmonic function theory and stationary dynamics, with potential consequences for random walk boundary theory and identification problems.
• Model Theory for Measurable Dynamics: The existence of universal compact Radon-Nikodym models advances the descriptive theory of measurable and nonsingular actions, offering a canonical topological setting for studying cocycle phenomena.
• Counterexamples and Limitations: The explicit non-$_\lambda$6 example underlines that classical regularity or geometric methods have intrinsic limitations in the lcsc group context—a caution for further generalizations.

Future avenues include the exploration of cocycle rigidity, explicit harmonic majorant computation in algebraic or Lie group settings, and further development of constructive and universal models for more general cocycle-valued dynamical invariants. Moreover, the constraints found in the locally compact setting motivate new techniques for non-discrete groups and for boundary theory in nonamenable and nonunimodular settings.

Conclusion

This work presents a comprehensive and rigorous development of the structure, regularity, and topological models of stationary actions for locally compact groups, centered on the analytic behavior of Radon-Nikodym cocycles and the geometric realization of stationary measures. The results extend and sharpen classical theorems, establish necessary and sufficient conditions for cocycle regularity, deliver new universal constructions, and produce explicit examples demonstrating the fundamental differences between the locally compact and discrete cases. The analytic machinery developed—particularly regarding the harmonic majorant and compact model construction—forms a technical foundation for future research in the operator algebra, ergodic theory, and stationary dynamics of locally compact groups.