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Existence and uniqueness of quasi-stationary and quasi-ergodic measures for absorbing Markov chains: a Banach lattice approach

Matheus M. Castro, Jeroen S. W. Lamb, Guillermo Olicón-Méndez, Martin Rasmussen

TL;DR

The paper develops a functional-analytic, Banach-lattice framework to establish existence and uniqueness of quasi-stationary and quasi-ergodic measures for absorbing Markov chains on general state spaces. By treating the transition kernel as a compact positive operator P on C^0(M) and analyzing its spectral structure, the authors identify a survival rate λ = r(P) and a finite peripheral spectrum {λ e^{2π i j/m}} with simple eigenvalues, yielding a positive eigenfunction f and a unique QSM μ. Under either m = 1 or ρ(Z) = 0, a unique QEM η is obtained via η(d x) = f(x) μ(d x) / ∫ f dμ, and the Yaglom limit and conditioned ergodic-type limits are characterized, including exponential convergence in the simple-peripheral-case. The theory covers cases with m > 1, where a cyclic decomposition still yields a unique QEM, and the results are illustrated through applications to random dynamical systems with bounded noise, broadening the scope beyond finite-state or strongly ergodic settings. Overall, the work provides weak, verifiable dynamical conditions under which quasi-stationary and quasi-ergodic behavior can be rigorously established for absorbing processes.

Abstract

We establish the existence and uniqueness of quasi-stationary and quasi-ergodic measures for almost surely absorbed discrete-time Markov chains under weak conditions. We obtain our results by exploiting Banach lattice properties of transition functions under natural regularity assumptions.

Existence and uniqueness of quasi-stationary and quasi-ergodic measures for absorbing Markov chains: a Banach lattice approach

TL;DR

The paper develops a functional-analytic, Banach-lattice framework to establish existence and uniqueness of quasi-stationary and quasi-ergodic measures for absorbing Markov chains on general state spaces. By treating the transition kernel as a compact positive operator P on C^0(M) and analyzing its spectral structure, the authors identify a survival rate λ = r(P) and a finite peripheral spectrum {λ e^{2π i j/m}} with simple eigenvalues, yielding a positive eigenfunction f and a unique QSM μ. Under either m = 1 or ρ(Z) = 0, a unique QEM η is obtained via η(d x) = f(x) μ(d x) / ∫ f dμ, and the Yaglom limit and conditioned ergodic-type limits are characterized, including exponential convergence in the simple-peripheral-case. The theory covers cases with m > 1, where a cyclic decomposition still yields a unique QEM, and the results are illustrated through applications to random dynamical systems with bounded noise, broadening the scope beyond finite-state or strongly ergodic settings. Overall, the work provides weak, verifiable dynamical conditions under which quasi-stationary and quasi-ergodic behavior can be rigorously established for absorbing processes.

Abstract

We establish the existence and uniqueness of quasi-stationary and quasi-ergodic measures for almost surely absorbed discrete-time Markov chains under weak conditions. We obtain our results by exploiting Banach lattice properties of transition functions under natural regularity assumptions.
Paper Structure (11 sections, 19 theorems, 94 equations)

This paper contains 11 sections, 19 theorems, 94 equations.

Table of Contents

  1. Introduction and motivation
  2. Main results
  3. Applications
  4. Some direct consequences of Hypothesis \ref{['(H)']}
  5. Banach lattices
  6. Banach lattice properties of P
  7. Existence and Uniqueness of a Quasi-stationary measures
  8. Existence of a Quasi-Ergodic Measure
  9. Analysis of the case m=1
  10. Analysis of the case m>1
  11. Proof of Lemma \ref{['Lobo']}

Key Result

Proposition 3.1

If $X_n$ fulfils Hypothesis (H), then the map $\mathcal{P}: L^\infty (M)\to L^\infty (M)$, $\mathcal{P}f= \mathbb E_x[f\circ X_1 ]=\int_M f(y) \mathcal{P}(x,\mathrm{d} y)$ has the following properties:

Theorems & Definitions (46)

  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Proposition 3.1
  • proof
  • ...and 36 more