Ergodicity of Poisson products and applications

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 $T_{\ast }$ which acts on realization of the Poisson process over $X$. We prove ergodicity of the Poisson-product $T\times T_{\ast }$ under the assumption that $T$ is ergodic and conservative. We then show, assuming ergodicity of $T\times T_{\ast }$, 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.

Published in at http://dx.doi.org/10.1214/12-AOP824 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• A stable marriage of Poisson and Lebesgue
• Poisson splitting by factors

Cited by in corpus (2)

• A monotone Sinai theorem
• Poisson-Orlicz norm and infinite Ergodic Theory