Ergodicity of Poisson products and applications

TL;DR

This paper investigates the ergodic properties of Poisson processes over measure spaces with measure-preserving transformations, establishing new results on ergodicity of Poisson products and implications for equivariant operations.

Contribution

It proves ergodicity of Poisson products under certain conditions and shows the impossibility of deterministic equivariant operations in this setting.

Findings

Proves ergodicity of Poisson-product $T\times T_*$ when $T$ is ergodic and conservative.

Demonstrates the impossibility of deterministic equivariant thinning, allocation, or matching.

Establishes ergodicity of the first return of left-most transformation and discusses spectral properties.

Abstract

In this paper we study the Poisson process over a $\sigma$-finite measure-space equipped with a measure preserving transformation or a group of measure preserving transformations. For a measure-preserving transformation $T$ acting on a $\sigma$-finite measure-space $X$, the Poisson suspension of $T$ is the associated probability preserving transformation $T_*$ which acts on realization of the Poisson process over $X$. We prove ergodicity of the Poisson-product $T\times T_*$ under the assumption that $T$ is ergodic and conservative. We then show, assuming ergodicity of $T\times T_*$, that it is impossible to deterministically perform natural equivariant operations: thinning, allocation or matching. In contrast, there are well-known results in the literature demonstrating the existence of isometry equivariant thinning, matching and allocation of homogenous Poisson processes on $\mathbb{R}^d$. We also prove ergodicity of the "first return of left-most transformation" associated with a measure preserving transformation on $\mathbb{R}_+$, and discuss ergodicity of the Poisson-product of measure preserving group actions, and related spectral properties.