The fractional Poisson process and other limit point processes for rare events in infinite ergodic theory

TL;DR

This paper investigates the limit behavior of return times to rare events in infinite-measure dynamical systems, revealing the emergence of fractional Poisson processes and characterizing their properties.

Contribution

It provides the first explicit identification of fractional Poisson processes in infinite ergodic theory and offers a general explanation for their appearance as fixed points of a functional equation.

Findings

Characterization of limit processes for all points in interval maps with a neutral fixed point

Identification of fractional (possibly compound) Poisson processes as limit processes

An abstract result explaining the emergence of fractional Poisson processes

Abstract

We study the process of suitably normalized successive return times to rare events in the setting of infinite-measure preserving dynamical systems. Specifically, we consider small neighborhoods of points whose measure tends to zero. We obtain two types of results. First, we conduct a detailed study of a class of interval maps with a neutral fixed point and we fully characterize the limit processes for all points, highlighting a trichotomy and the emergence of the fractional (possibly compound) Poisson process. This is the first time that these processes have been explicitly identified in this context. Second, we prove an abstract result that offers an explanation for the emergence of the fractional Poisson process, as the unique fixed point of a functional equation, drawing a parallel with the well-established behavior of the Poisson process in finite-measure preserving dynamical systems.