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Jumping for diffusion in random metastable systems

Cecilia González-Tokman, Joshua Peters

TL;DR

The paper develops a rigorous framework for random metastable interval maps, where small random perturbations introduce holes that couple multiple invariant subintervals. It establishes a spectral theory for the open Perron-Frobenius cocycle, proving a quasi-compact, hyperbolic structure and a precise eigenpair behavior under perturbation, and shows that the jump dynamics of the perturbed system converge to those of an averaged Markov jump process with generator $\bar{G}$. It then connects these jumps to a quenched Central Limit Theorem for fibrewise observables and derives a diffusion-coefficient formula in terms of the averaged generator, providing a concrete link between metastable dynamics and averaged Markov dynamics. The theoretical results are illustrated with an application to random paired tent maps, yielding explicit expressions for the diffusion limit in terms of averaged leakage rates. Overall, the work integrates random dynamical systems, transfer-operator perturbation theory, and Markov-jump approximations to quantify metastable fluctuations in random environments, with potential implications for understanding diffusion-like behaviour in complex, randomly forced systems.

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.

Jumping for diffusion in random metastable systems

TL;DR

The paper develops a rigorous framework for random metastable interval maps, where small random perturbations introduce holes that couple multiple invariant subintervals. It establishes a spectral theory for the open Perron-Frobenius cocycle, proving a quasi-compact, hyperbolic structure and a precise eigenpair behavior under perturbation, and shows that the jump dynamics of the perturbed system converge to those of an averaged Markov jump process with generator . It then connects these jumps to a quenched Central Limit Theorem for fibrewise observables and derives a diffusion-coefficient formula in terms of the averaged generator, providing a concrete link between metastable dynamics and averaged Markov dynamics. The theoretical results are illustrated with an application to random paired tent maps, yielding explicit expressions for the diffusion limit in terms of averaged leakage rates. Overall, the work integrates random dynamical systems, transfer-operator perturbation theory, and Markov-jump approximations to quantify metastable fluctuations in random environments, with potential implications for understanding diffusion-like behaviour in complex, randomly forced systems.

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.

Paper Structure

This paper contains 12 sections, 25 theorems, 205 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Preliminaries
  4. Metastable systems and their perturbations
  5. The initial system
  6. The perturbations
  7. The open system
  8. The jump processes
  9. The averaged Markov jump process
  10. Approximation of jumps for random metastable systems
  11. The diffusion coefficient
  12. Example: Random paired tent maps

Key Result

Theorem 1.1

Let $\{(\Omega,\mathcal{F},\mathbb{P},\sigma,\mathop{\mathrm{BV}}\limits(I),\mathcal{L}^\varepsilon)\}_{\varepsilon\geq 0}$ be a sequence of random dynamical systems of metastable maps $T_\omega^\varepsilon:I\to I$ satisfying list:I1-list:I6 and list:P1-list:P7. For each $j\in \{1,\cdots,m\}$, if $\ Further, and,

Figures (1)

  • Figure 1: Paired tent map $T_{a,b}$ on $I=[-1,1]$ with $a=b=0$ (left) and $a=b=\frac{1}{10}$ (right)

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7: Perron-Frobenius operator cocycle
  • ...and 89 more