Perturbation theory for killed Markov processes and quasi-stationary distributions

We investigate the stability of quasi-stationary distributions of killed Markov processes to perturbations of the generator. In the first setting, we consider a general bounded self-adjoint perturbation operator, and after that, study a particular unbounded perturbation corresponding to the truncation of the killing rate. In both scenarios, we quantify the difference between eigenfunctions of the smallest eigenvalue of the perturbed and unperturbed generator in a Hilbert space norm. As a consequence, $\mathcal{L}^1$-norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided. These results are particularly relevant to the recently-proposed class of quasi-stationary Monte Carlo methods, designed for scalable exact Bayesian inference.