Dynamical systems with heavy-tailed random parameters

TL;DR

This paper investigates the long-term behavior of random dynamical systems with heavy-tailed parameters, providing classification and recurrence conditions without requiring ellipticity, using Markov chain and Lyapunov function techniques.

Contribution

It introduces a classification framework for such systems and sharp recurrence criteria based on passage time moments, expanding understanding of their asymptotic behavior.

Findings

Classified systems according to their recurrence properties

Established sharp conditions for passage time moments

Mapped systems to Markov chains with heavy-tailed innovations

Abstract

Motivated by the study of the time evolution of random dynamical systems arising in a vast variety of domains --- ranging from physics to ecology ---, we establish conditions for the occurrence of a non-trivial asymptotic behaviour for these systems in the absence of an ellipticity condition. More precisely, we classify these systems according to their type and --- in the recurrent case --- provide with sharp conditions quantifying the nature of recurrence by establishing which moments of passage times exist and which do not exist. The problem is tackled by mapping the random dynamical systems into Markov chains on $\mathbb{R}$ with heavy-tailed innovation and then using powerful methods stemming from Lyapunov functions to map the resulting Markov chains into positive semi-martingales.