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

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 $r_{ess}(P)\le r_{ess}(Q)$ whenever $0\le P\le Q$ as kernels, a property that is known to fail in general on $L^p$ spaces, $p<+\mathrm{\infty }$. 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.