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Quasi-compactness for dominated kernels with application to quasi-stationary distribution theory

Denis Villemonais

TL;DR

This paper develops a flexible domination framework to bound the essential spectral radius of positive operators and derives practical Lyapunov‑type criteria for quasi‑compactness on weighted $L^ obreakspace^ obreakspace^ obreakspace spaces. It then characterizes the asymptotics of iterates of quasi‑compact kernels and the long‑time behavior of quasi‑compact semigroups, including non‑conservative settings. The results are applied to absorbed Markov processes to establish existence and convergence to quasi‑stationary distributions under domination by regular kernels or locally integrable densities, relaxing strong Feller regularity. Overall, the work unifies and extends quasi‑compactness theory for non‑negative operators and provides tools for QSD in reducible and general state spaces.

Abstract

We establish a domination principle for positive operators, which provides an upper bound on the essential spectral radius and yields quasi-compactness criteria on weighted supremum spaces with Lyapunov type functions and local domination. In particular, for kernels acting on such spaces, we obtain $r_{ess}(P)\leq r_{ess}(Q)$ whenever $0\leq P\leq Q$ as kernels, a property that is known to fail in general on $L^p$ spaces, $p<+\infty$. We then describe the asymptotics of iterates of positive quasi-compact kernels, showing convergence, after suitable renormalization, towards a finite decomposition over eigenelements, and we study the long-time behaviour of quasi-compact continuous-time semigroups. For the latter, we prove that measurability in time and quasi-compactness at a single positive time imply quasi-compactness at all times, exclude periodic behaviour, and entail convergence to eigenelements as time goes to infinity. Finally, we apply these results to absorbed Markov processes and quasi-stationary distributions. In this setting, the domination and Lyapunov criteria allow one to work in reducible situations and to relax classical regularity assumptions, for instance replacing strong Feller conditions by domination from a regular kernel or by locally uniformly integrable densities on suitable weighted supremum spaces.

Quasi-compactness for dominated kernels with application to quasi-stationary distribution theory

TL;DR

This paper develops a flexible domination framework to bound the essential spectral radius of positive operators and derives practical Lyapunov‑type criteria for quasi‑compactness on weighted $L^ obreakspace^ obreakspace^ obreakspace spaces. It then characterizes the asymptotics of iterates of quasi‑compact kernels and the long‑time behavior of quasi‑compact semigroups, including non‑conservative settings. The results are applied to absorbed Markov processes to establish existence and convergence to quasi‑stationary distributions under domination by regular kernels or locally integrable densities, relaxing strong Feller regularity. Overall, the work unifies and extends quasi‑compactness theory for non‑negative operators and provides tools for QSD in reducible and general state spaces.

Abstract

We establish a domination principle for positive operators, which provides an upper bound on the essential spectral radius and yields quasi-compactness criteria on weighted supremum spaces with Lyapunov type functions and local domination. In particular, for kernels acting on such spaces, we obtain whenever as kernels, a property that is known to fail in general on spaces, . We then describe the asymptotics of iterates of positive quasi-compact kernels, showing convergence, after suitable renormalization, towards a finite decomposition over eigenelements, and we study the long-time behaviour of quasi-compact continuous-time semigroups. For the latter, we prove that measurability in time and quasi-compactness at a single positive time imply quasi-compactness at all times, exclude periodic behaviour, and entail convergence to eigenelements as time goes to infinity. Finally, we apply these results to absorbed Markov processes and quasi-stationary distributions. In this setting, the domination and Lyapunov criteria allow one to work in reducible situations and to relax classical regularity assumptions, for instance replacing strong Feller conditions by domination from a regular kernel or by locally uniformly integrable densities on suitable weighted supremum spaces.
Paper Structure (10 sections, 15 theorems, 74 equations)

This paper contains 10 sections, 15 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Quasi-compactness by domination arguments
  3. Lyapunov type criterion for quasi-compactness in weighted spaces
  4. Asymptotics of iterates of quasi-compact kernels and of quasi-compact semigroups
  5. Iterates of quasi-compact kernels
  6. Continuous time semigroups
  7. Applications to quasi-stationary distributions
  8. Lazy Markov chain with killing
  9. Kernel with locally uniformly integrable densities
  10. Sub-Markov processes dominated by a strong Feller kernel

Key Result

Theorem 1

Let $P$ be a positive operator on $B$ and assume that there exists two positive operators $K$ and $S$ such that $K$ is compact, $(P-S)_+$ is well defined and $0\leq P\leq K+S$. Then $r_{ess}(P)\leq r(S)$.

Theorems & Definitions (42)

  • Remark 1
  • Theorem 1
  • Corollary 2
  • proof : Proof of Theorem \ref{['thm:QCbyDOM']}
  • Lemma 3
  • proof : Proof of Lemma \ref{['lem:compactproduct']}
  • Corollary 4
  • proof : Proof of Corollary \ref{['cor:BbE']}
  • Remark 2
  • Proposition 5
  • ...and 32 more