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Averaging for random metastable systems

Cecilia González-Tokman, Joshua Peters

TL;DR

This work analyzes random metastable dynamics generated by small perturbations of a deterministic, two-interval expanding map. It proves that the invariant density of the perturbed system converges in $L^1$ to a non-random convex combination of the two original invariant densities, with weights determined by spatial averages of the perturbation holes. It also characterizes the limiting second Oseledets space, yielding a one-dimensional coherent structure that behaves slowly as perturbations vanish. The framework is demonstrated on random paired tent maps and extended to systems with $m\ge 2$ initial invariant sets, linking the infinite-dimensional dynamics to finite-state Markov chains in random environments and providing explicit formulas for limiting densities and weights.

Abstract

Random metastability occurs when an externally forced or noisy system possesses more than one state of apparent equilibrium. This work investigates a class of random dynamical systems, arising from perturbing a one-dimensional piecewise smooth expanding map of the interval with two invariant subintervals, each supporting a unique ergodic absolutely continuous invariant measure. Upon perturbation, this invariance is destroyed, allowing trajectories to randomly switch between subintervals. We show that the invariant density of the randomly perturbed system may be approximated by an explicit convex combination of the two initially invariant densities, obtained by averaging. Further, we also identify the limit of the second Oseledets space, or coherent structure, as the perturbation shrinks to zero. Our results are applied to random paired tent maps over ergodic, measure-preserving, and invertible driving systems. Finally, we provide generalisations to systems admitting more than two initially invariant sets.

Averaging for random metastable systems

TL;DR

This work analyzes random metastable dynamics generated by small perturbations of a deterministic, two-interval expanding map. It proves that the invariant density of the perturbed system converges in to a non-random convex combination of the two original invariant densities, with weights determined by spatial averages of the perturbation holes. It also characterizes the limiting second Oseledets space, yielding a one-dimensional coherent structure that behaves slowly as perturbations vanish. The framework is demonstrated on random paired tent maps and extended to systems with initial invariant sets, linking the infinite-dimensional dynamics to finite-state Markov chains in random environments and providing explicit formulas for limiting densities and weights.

Abstract

Random metastability occurs when an externally forced or noisy system possesses more than one state of apparent equilibrium. This work investigates a class of random dynamical systems, arising from perturbing a one-dimensional piecewise smooth expanding map of the interval with two invariant subintervals, each supporting a unique ergodic absolutely continuous invariant measure. Upon perturbation, this invariance is destroyed, allowing trajectories to randomly switch between subintervals. We show that the invariant density of the randomly perturbed system may be approximated by an explicit convex combination of the two initially invariant densities, obtained by averaging. Further, we also identify the limit of the second Oseledets space, or coherent structure, as the perturbation shrinks to zero. Our results are applied to random paired tent maps over ergodic, measure-preserving, and invertible driving systems. Finally, we provide generalisations to systems admitting more than two initially invariant sets.
Paper Structure (16 sections, 25 theorems, 140 equations, 3 figures)

This paper contains 16 sections, 25 theorems, 140 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Preliminaries
  4. Paired metastable systems and their perturbations
  5. The initial system
  6. The perturbations
  7. Continuity of Oseledets Decomposition
  8. Characterisation of the limiting invariant density
  9. The weights
  10. The limiting invariant density
  11. Characterisation of the second Oseledets space
  12. The limiting invariant density for multiple initially invariant states
  13. Example: Random paired tent maps
  14. The system of concern
  15. Conditions for the unperturbed map
  16. ...and 1 more sections

Key Result

Theorem A

Let $\{(\Omega,\mathcal{F},\mathbb{P},\sigma,\mathop{\mathrm{BV}}\limits([-1,1]),\mathcal{L}^\varepsilon)\}_{\varepsilon\geq 0}$ be a family of random dynamical systems of paired metastable maps $T_\omega^\varepsilon:[-1,1]\to [-1,1]$ satisfying list:I1-list:I6 and list:P1-list:P7 (see Section sec:m uniformly over $\omega\in\Omega$ away from a $\mathbb{P}$-null set.

Figures (3)

  • Figure 1: Transition probabilities between $S_L$ and $S_R$ for a fixed $\omega\in\Omega$.
  • Figure 2: Splitting of $E_{t,\omega}^\varepsilon$
  • Figure 3: Paired tent map $T_{a,b}$ on $I=[-1,1]$ with $a=b=0$ (left) and $a=b=\frac{1}{10}$ (right)

Theorems & Definitions (100)

  • Theorem A
  • Theorem B
  • Theorem C
  • 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 90 more