ERGODICITY OF POISSON PRODUCTS AND APPLICATIONS

作者：Meyerovitch Tom^*

来源：Annals of Probability, 2013, 41(5): 3181-3200.

DOI：10.1214/12-AOP824

• Poisson suspension
• equivariant thinning
• equivariant allocation
• infinite measure preserving transformations
• conservative transformations

In this paper we study the Poisson process over a a-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 a-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 x T-* under the assumption that T is ergodic and conservative. We then show, assuming ergodicity of T x 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 R-d. We also prove ergodicity of the %26quot;first return of left-most transformation%26quot; associated with a measure preserving transformation on R+, and discuss ergodicity of the Poisson-product of measure preserving group actions, and related spectral properties.

• 出版日期2013-9