Decay of correlations and laws of rare events for transitive random maps

TL;DR

This paper demonstrates that random perturbations of transitive maps lead to exponential decay of correlations and establish the limiting distributions for extreme events and hitting times as standard exponential or Poisson processes.

Contribution

It proves that random perturbations of transitive maps satisfy conditions for decay of correlations and characterizes the limiting distributions for rare events and hitting times.

Findings

Exponential decay of correlations for perturbed transitive maps

Limiting distributions for EVLs and HTS/RTS are standard exponential

REPP converges to a standard Poisson process

Abstract

We show that a uniformly continuous random perturbation of a transitive map defines an aperiodic Harris chain which also satisfies Doeblin's condition. As a result, we get exponential decay of correlations for suitable random perturbations of such systems. We also prove that, for transitive maps, the limiting distribution for Extreme Value Laws (EVLs) and Hitting/Return Time Statistics (HTS/RTS) is standard exponential. Moreover, we show that the Rare Event Point Process (REPP) converges in distribution to a standard Poisson process.