Jumping for diffusion in random metastable systems

TL;DR

This paper studies metastable behavior in randomly perturbed dynamical systems, showing that Markov chain jump distributions approximate system jumps and diffusion coefficients, with applications to random tent maps.

Contribution

It introduces a method to approximate metastable system jumps and diffusion behavior using Markov chains, extending understanding of random perturbations in dynamical systems.

Findings

Markov chain jump distributions approximate metastable system jumps

Diffusion coefficients of observables are similarly approximated

Results applied to random paired tent maps

Abstract

Random metastability occurs when an externally forced or noisy system possesses more than one state of apparent equilibrium. This work investigates fluctuations in a class of random dynamical systems, arising from randomly perturbing a piecewise smooth expanding interval map with more than one invariant subinterval. Upon perturbation, this invariance is destroyed, allowing trajectories to switch between subintervals, giving rise to metastable behaviour. We show that the distributions of jumps of a time-homogeneous Markov chain approximate the distributions of jumps for random metastable systems. Additionally, we demonstrate that this approximation extends to the diffusion coefficient for (random) observables of such systems. As an example, our results are applied to Horan's random paired tent maps.