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

2024/11/28 by Dylan Bansard-Tresse, Bansard-Tresse, Dylan
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2411.19337

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.

Related

• Recurrence to rare events for all points in countable Markov shifts
• Hitting Times and Positions in Rare Events
• Rare event process and entry times distribution for arbitrary null sets on compact manifolds
• Spatio-temporal Poisson processes for visits to small sets
• Strong memoryless times and rare events in Markov renewal point processes