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

TL;DR

This paper develops a domination principle for positive operators, establishing quasi-compactness criteria on weighted supremum spaces, and applies these results to analyze the long-term behavior of absorbed Markov processes and quasi-stationary distributions.

Contribution

It introduces a new domination principle for positive operators, providing quasi-compactness criteria applicable to a broad class of kernels and processes, especially in reducible and less regular settings.

Findings

Established a domination principle for positive operators.

Derived quasi-compactness criteria on weighted supremum spaces.

Analyzed asymptotics and convergence of quasi-compact kernels and semigroups.

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.