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Perturbation theory for killed Markov processes and quasi-stationary distributions

Part of: Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  19 March 2025

Daniel Rudolf  and
Andi Q. Wang
Daniel Rudolf*
Affiliation:
Universität Passau
Andi Q. Wang*
Affiliation:
University of Warwick
*
*Postal address: Faculty of Computer Science and Mathematics, Universität Passau, Innstrasse 33, 94032 Passau, Germany. Email: daniel.rudolf@uni-passau.de
**Postal address: Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. Email: andi.wang@warwick.ac.uk
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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 then 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, $\mathcal{L}^1$-norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided.

Keywords

Monte Carlo methods

MSC classification

Primary: 60J35: Transition functions, generators and resolvents
Secondary: 65C05: Monte Carlo methods 60J22: Computational methods in Markov chains

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 57 , Issue 3 , September 2025 , pp. 1138 - 1165
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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