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Metastability of random maps: a resolvent approach

Diego Marcondes, Sandro Vaienti

TL;DR

The paper develops a resolvent-based framework to analyze metastability in randomly perturbed one-dimensional maps with uncountable state spaces, by integrating Markov-process resolvent methods with spectral analysis of transfer operators. Metastability is characterized through speeded-up order processes and a reduced generator on metastable wells, with sufficient conditions ${\mathfrak R}^{(1)}$ and ${\mathfrak R}^{(2)}_{\mathcal{L}}$ ensuring convergence to a limiting Markov process. It proves stochastic stability: the stationary measure $\mu_{\varepsilon}$ converges strongly to a convex combination of invariant measures of the unperturbed map, with explicit rates derived via Chen–Stein methods. The framework is illustrated with expanding maps having two or three wells, showing that metastable dynamics and explicit rates can be obtained even when perturbations are uncountably many and the state space is continuous, opening avenues for higher-dimensional and more general hyperbolic systems.

Abstract

We present a general framework to study the metastability of random perturbations of dynamical systems. It integrates techniques from the theory of Markov processes, in particular the resolvent approach to metastability, with the spectral analysis of transfer operators associated to the dynamics. The proposed framework is applied to study the metastability of one-dimensional dynamical systems generated by a map randomly perturbed by sub-Gaussian noise.

Metastability of random maps: a resolvent approach

TL;DR

The paper develops a resolvent-based framework to analyze metastability in randomly perturbed one-dimensional maps with uncountable state spaces, by integrating Markov-process resolvent methods with spectral analysis of transfer operators. Metastability is characterized through speeded-up order processes and a reduced generator on metastable wells, with sufficient conditions and ensuring convergence to a limiting Markov process. It proves stochastic stability: the stationary measure converges strongly to a convex combination of invariant measures of the unperturbed map, with explicit rates derived via Chen–Stein methods. The framework is illustrated with expanding maps having two or three wells, showing that metastable dynamics and explicit rates can be obtained even when perturbations are uncountably many and the state space is continuous, opening avenues for higher-dimensional and more general hyperbolic systems.

Abstract

We present a general framework to study the metastability of random perturbations of dynamical systems. It integrates techniques from the theory of Markov processes, in particular the resolvent approach to metastability, with the spectral analysis of transfer operators associated to the dynamics. The proposed framework is applied to study the metastability of one-dimensional dynamical systems generated by a map randomly perturbed by sub-Gaussian noise.
Paper Structure (33 sections, 19 theorems, 230 equations, 5 figures)

This paper contains 33 sections, 19 theorems, 230 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Metastability of Markov processes
  3. Metastability of random perturbations of a map
  4. A resolvent approach to the metastability of random maps
  5. Paper structure
  6. Notation
  7. A resolvent approach to the metastability of randomly perturbed maps
  8. Markov chain associated with randomly perturbed maps
  9. Markov process associated with randomly perturbed maps
  10. Metastable wells
  11. Speeded-up and order processes
  12. Metastability
  13. Resolvent approach to metastability
  14. Sufficient conditions for the metastability of random mixing maps
  15. A sufficient condition for $\mathfrak{R}^{(1)}$
  16. ...and 18 more sections

Key Result

Proposition 2.1

Fix $\epsilon > 0$. If there exist $n_{\epsilon} < \infty$ and $A_{\epsilon} \subset I$, with $\text{Leb}(A_{\epsilon}) > 0$ and $\rho_{\epsilon}(x,y) = \rho_{\epsilon}(x,y) \chi_{A_{\epsilon}}(y)$ for all $x,y \in I$, such that then $X_{n}^{\epsilon}$ has a unique ACIM.

Figures (5)

  • Figure 1: Example of a map $T$ that satisfies (A1) with an illustration of the holes and metastable wells.
  • Figure 2: Map $\tilde{T}^{1}_{\epsilon}(\omega,\cdot)$ in red for fixed $\omega \in \Omega$ with $I_{1} = (0,1/2)$ when $T$ is the map in Figure \ref{['f1']} and $T_{\epsilon}(\omega,\cdot) = T(\cdot) + \sigma(\omega)$ is in black with $\sigma(\omega) > 0$. Observe that the maps $\tilde{T}^{1}_{\epsilon}(\omega,\cdot)$ and $T_{\epsilon}(\omega,\cdot)$ coincide for $x$ satisfying $T_{\epsilon}(\omega,x) \in (0,1/2)$.
  • Figure 3: Examples of maps $T$ for which the developed theory could be applied.
  • Figure 4: Example of a map $T$ that satisfies (A1) with an illustration of the holes and metastable wells.
  • Figure 5: Map $\tilde{T}^{i}_{\epsilon}(\omega,\cdot)$ for $i = 2$ in red for fixed $\omega \in \Omega$ with $I_{2} = (1/3,2/3)$ when $T$ is the map in Figure \ref{['f4']}. The perturbed map satisfies $T_{\epsilon}(\omega,x) = T(x) + \sigma(\omega)$ for $x \in (1/3,\bar{x})$ and $T_{\epsilon}(\omega,x) = T(x) - \sigma(\omega)$ for $x \in (\bar{x},2/3)$, and it is represented in black for $\sigma(\omega) < 0$ in (A) and for $\sigma(\omega) > 0$(B). Observe that the maps $\tilde{T}^{i}_{\epsilon}(\omega,\cdot)$ and $T_{\epsilon}(\omega,\cdot)$ coincide for $x$ satisfying $T_{\epsilon}(\omega,x) \in (1/3,2/3)$.

Theorems & Definitions (42)

  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Remark 3.6
  • ...and 32 more