Perturbation theory for killed Markov processes and quasi-stationary distributions

TL;DR

This paper analyzes how small changes in the generator of killed Markov processes affect their quasi-stationary distributions, providing bounds on the differences in eigenfunctions and distributions.

Contribution

It introduces a perturbation framework for killed Markov processes, quantifying the stability of quasi-stationary distributions under generator perturbations.

Findings

Eigenfunction differences are bounded in a Hilbert space norm.

L1 norm estimates relate distribution differences to generator perturbations.

Results apply to both bounded and unbounded perturbations.

Abstract

Motivated by recent developments of quasi-stationary Monte Carlo methods, we investigate the stability of quasi-stationary distributions of killed Markov processes under perturbations of the generator. We first consider a general bounded self-adjoint perturbation operator, and after that, study a particular unbounded perturbation corresponding to truncation of the killing rate. In both scenarios, we quantify the difference between eigenfunctions of the smallest eigenvalue of the perturbed and unperturbed generators in a Hilbert space norm. As a consequence, L1 norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided.