November 2023 Isomorphisms of Poisson systems over locally compact groups
Amanda Wilkens
Author Affiliations +
Ann. Probab. 51(6): 2158-2191 (November 2023). DOI: 10.1214/23-AOP1642


A Poisson system is a Poisson point process and a group action, together forming a measure-preserving dynamical system. Ornstein and Weiss proved Poisson systems over many amenable groups were isomorphic in their 1987 paper. We consider Poisson systems over nondiscrete, noncompact, locally compact Polish groups, and we prove by construction all Poisson systems over such a group are finitarily isomorphic, producing examples of isomorphisms for nonamenable group actions. As a corollary, we prove Poisson systems and products of Poisson systems are finitarily isomorphic.

For a Poisson system over a group belonging to a slightly more restrictive class than above, we further prove it splits into two Poisson systems whose intensities sum to the intensity of the original, generalizing the same result for Poisson systems over Euclidean space proved by Holroyd, Lyons and Soo.

Funding Statement

This research was supported in part by NSF Grant DMS-1937215.


For my brother, Daniel, and in memory of him.


The author thanks Lewis Bowen for helpful comments and discussions provided throughout the writing and revising of this paper, particularly those on word metrics, coinduction and generating sets of compact groups, and Brandon Seward for inspiring the project by suggesting the construction in [22] might work for Lie groups.

The author also thanks the referee for their many facilitative and insightful comments.


Download Citation

Amanda Wilkens. "Isomorphisms of Poisson systems over locally compact groups." Ann. Probab. 51 (6) 2158 - 2191, November 2023.


Received: 1 December 2021; Revised: 1 May 2023; Published: November 2023
First available in Project Euclid: 12 November 2023

Digital Object Identifier: 10.1214/23-AOP1642

Primary: 37A35
Secondary: 60G10 , 60G55

Keywords: finitary isomorphism , isomorphism , locally compact second countable , nonamenable , Poisson point process , Poisson splitting , Poisson system

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 6 • November 2023
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