Stationary Actions in Locally Compact Groups

• Stationary actions are defined for locally compact groups via the condition µ * ν = ν, generalizing classical invariant measures.
• Analytic tools such as Radon–Nikodym cocycles and harmonic majorants are employed to control measure regularity and explore rigidity phenomena.
• The framework unveils structural decompositions akin to the Furstenberg–Zimmer theorem, offering insights into ergodic types and nonamenable dynamics.

A stationary action of a locally compact group encapsulates the interplay between group dynamics, random walks, measure theory, and boundary theory on noncompact spaces. For a locally compact second-countable group $G$ endowed with a probability measure $\mu$, a $G$-space $(X,\nu)$ is $\mu$-stationary if $\mu * \nu = \nu$, meaning $\nu$ is preserved in mean under the random walk generated by $\mu$. Stationary measures provide a framework that generalizes invariant measures and unlocks deep structural, rigidity, and analytic properties of group actions, particularly in contexts lacking $G$-invariance.

1. Definitions and Foundational Structures

Given a locally compact second-countable (lcsc) group $G$ and a Borel probability measure $\mu$0 that is admissible (absolutely continuous with respect to Haar measure and whose support generates $\mu$1 as a closed semigroup), a $\mu$2-space is a standard Borel space $\mu$3 with a Borel $\mu$4-action. A Borel probability measure $\mu$5 is quasi-invariant if $\mu$6 for all $\mu$7.

A $\mu$8-space $\mu$9 is $G$0-stationary if, for every Borel set $G$1,

$G$2

Equivalently, for $G$3-almost every $G$4, $G$5. The family of Radon–Nikodym derivatives $G$6 forms a measurable cocycle, satisfying $G$7 $G$8-almost everywhere and, when possible, pointwise for strict versions (Avraham-Re'em et al., 10 Apr 2026).

In the topological context, if $G$9 is a finitely generated group and $(X,\nu)$0 is nondegenerate and finitely supported, a Radon measure $(X,\nu)$1 (assigning finite mass to compact sets) on a locally compact Hausdorff $(X,\nu)$2 is $(X,\nu)$3-stationary as above when the mean-value equation holds for all $(X,\nu)$4 (Alhalimi et al., 2024).

2. Existence of Stationary Measures and the Stationary Tarski Theorem

A key existence result is that every co-compact action of a finitely generated group $(X,\nu)$5 on a locally compact Hausdorff space admits a nonzero $(X,\nu)$6-stationary Radon measure, regardless of amenability. Explicitly, if $(X,\nu)$7 acts co-compactly on $(X,\nu)$8 (i.e., there exists compact $(X,\nu)$9 with $\mu$0), then there is a nonzero Radon measure $\mu$1 satisfying $\mu$2 (Alhalimi et al., 2024).

The existence is proven via a stationary analogue of Tarski's theorem: for every nonempty subset $\mu$3, there exists a finitely additive, $\mu$4-stationary measure $\mu$5 on $\mu$6 with $\mu$7 and $\mu$8 for all $\mu$9. The construction is built on potential theory, using the Green function $\mu * \nu = \nu$0. Two disjoint cases are considered based on the total Green mass of $\mu * \nu = \nu$1 (finite or infinite), and in both situations, compactness arguments and the property of stationarity are pivotal.

A nonzero $\mu * \nu = \nu$2-stationary Radon measure is then constructed by pulling back $\mu * \nu = \nu$3 functions to $\mu * \nu = \nu$4, using the finitely additive measure to generate a positive stationary linear functional, and invoking the Riesz–Markov representation theorem (Alhalimi et al., 2024).

This result demonstrates that in the nonamenable setting, where no invariant Radon measure may exist, a nonzero infinite stationary measure is nonetheless always present for co-compact actions. The distinction between invariant and stationary measures is thus crucial: amenability is equivalent to the existence of invariant measures on all compact actions, but stationarity always holds for co-compact actions.

3. Structure Theory and the Furstenberg–Zimmer Analogue

Stationary actions of lcsc groups admit a deep structural decomposition paralleling—yet generalizing—the classical Furstenberg–Zimmer structure theorem. For any $\mu * \nu = \nu$5-stationary action $\mu * \nu = \nu$6, there exists a diagram of factors with

• $\mu * \nu = \nu$7 such that $\mu * \nu = \nu$8 is a weakly mixing extension and $\mu * \nu = \nu$9 is a distal, measure-preserving action (Edeko, 2022).

The proof centers on random-walk (Markov) operators and conditional $\nu$0-modules ($\nu$1), with the dichotomy that any stationary extension is either weakly mixing or admits a nontrivial isometric (distal) intermediate factor. The construction uses transfinite recursion over intermediate factors, with separability arguments ensuring the tower terminates (Edeko, 2022).

In this framework, new phenomena absent in strictly measure-preserving dynamics arise: stationary actions may lack invariant measures but still admit a distal factor and a weakly mixing “noise” extension; the distal factor remains measure-preserving even if the ambient system does not.

Illustrative examples include stationary measures for projective group actions and the structure of the Poisson boundary, which is always a weakly mixing extension of a measure-preserving distal system (Edeko, 2022).

4. Classification, Rigidity, and Cocycles

The dynamical types of stationary actions are sharply constrained compared to nonsingular actions. The following rigidity results hold (Avraham-Re'em et al., 10 Apr 2026):

• Every stationary action of a noncompact group is conservative: for every Borel set $\nu$2 of positive measure, the set $\nu$3 has infinite Haar measure.
• Stationary actions are never of type I (purely dissipative) or type II$\nu$4 (infinite invariant measure); ergodic stationary actions are either of type II$\nu$5 (invariant probability measure) or type III (no σ-finite invariant measure).
• For any stationary action of type III$\nu$6 (maximal type III), it is possible to construct stationary actions of every type III$\nu$7 via skew-products and Maharam extensions, encoding all possible ratio sets.

The Radon–Nikodym cocycle $\nu$8 attached to a stationary action satisfies an almost-cocycle equation and encodes the full measured orbit equivalence class. Analytic control of $\nu$9 is given via positive $\mu$0-harmonic functions:

$\mu$1

with the harmonic majorant $\mu$2 controlling the oscillation of $\mu$3. Harnack-type inequalities hold where $\mu$4 is locally bounded, which is guaranteed when $\mu$5 has compact support and admits an $\mu$6 density, ensuring locally uniform regularity of $\mu$7 (Avraham-Re'em et al., 10 Apr 2026).

5. Universal Models and Regularity Phenomena

Every lcsc $\mu$8 admits a universal compact $\mu$9-space $G$0 (Mackey–Varadarajan model) into which arbitrary Borel $G$1-spaces embed equivariantly. Under analytic control assumptions (e.g., $G$2), there exists a universal compact model $G$3 with a continuous $G$4-harmonic cocycle $G$5 such that every stationary action embeds into $G$6 carrying its Radon–Nikodym cocycle to $G$7. The construction uses the compactness and equicontinuity of the cone of positive $G$8-harmonic functions via Arzelà–Ascoli, with an explicit realization of the cocycle structure (Avraham-Re'em et al., 10 Apr 2026).

However, the affine group demonstrates the failure of universal regularity. For a random walk on $G$9 in the contracting regime, the Poisson kernel (Radon–Nikodym derivative) can be unbounded near the identity, violating Harnack’s inequality and the SAT* property. No continuous compact model can realize the Poisson boundary cocycle in this setting, indicating essential limitations in the theory and reflecting intricate interactions between analytic and probabilistic features of stationary boundaries (Avraham-Re'em et al., 10 Apr 2026).

6. Connections and Further Directions

Connections to random walks and boundary theory are central. The construction of stationary measures recalls methods from Martin boundary theory, with the Green function playing a prominent role. The Poisson boundary $G$0 of a measured group is always a weakly mixing extension in the stationary category, with no nontrivial isometric factors (Björklund’s criterion) (Edeko, 2022). Analytic tools such as the harmonic majorant provide direct control over possible cocycle behaviors, and potential-theoretic methods inform possible generalizations beyond finitely supported measures (Alhalimi et al., 2024).

Further expected extensions include analogues for more general measured groups (e.g., spread-out or absolutely continuous measures), and for actions of more general groups (beyond countable or discrete). The robust invariance versus stationarity dichotomy highlights the nuanced differences in their ergodic/decomposition theories: amenability characterizes invariance, while co-compactness suffices for nonzero stationary measures.

Counterexamples and rigidity phenomena, such as the absence of stationary measures in certain non-co-compact settings or the unboundedness of Poisson kernels, illustrate both the universality and limitations of stationary measure theory for lcsc group actions (Alhalimi et al., 2024, Avraham-Re'em et al., 10 Apr 2026).