Poisson suspensions and Sushis

TL;DR

This paper demonstrates that certain ergodic point processes driven by infinite measure transformations are superpositions of shifted Poisson processes, leading to significant implications for their joinings and disjointness properties.

Contribution

It establishes a rigidity result showing ergodic point processes with all moments are superpositions of shifted Poisson processes, extending understanding of Poisson suspensions.

Findings

Ergodic point processes with all moments are superpositions of shifted Poisson processes.

Poisson suspension's ergodic self-joinings are Poisson joinings.

Provides an analog of the GAG property for Gaussian systems in the Poisson context.

Abstract

In this paper, we prove that ergodic point processes with moments of all orders, driven by particular infinite measure preserving transformations, have to be a superposition of shifted Poisson processes. This rigidity result has a lot of implications in terms of joining and disjointness for the corresponding Poisson suspension. In particular, we prove that its ergodic self-joinings are Poisson joinings, which provides an analog, in the Poissonian context, of the GAG property for Gaussian dynamical systems.