Simulation of quasi-stationary distributions on countable spaces

TL;DR

This paper reviews methods for simulating quasi-stationary distributions in countable spaces, focusing on Fleming-Viot dynamics, and provides new proofs and extensions of existing results.

Contribution

It introduces new proofs and extensions for the approximation methods of QSDs, especially in the context of Fleming-Viot dynamics, enhancing simulation techniques.

Findings

Comparison of different QSD approximation methods

Extension of known results with new proofs

Application of Fleming-Viot dynamics for simulation

Abstract

Quasi-stationary distributions (QSD) have been widely studied since the pioneering work of Kolmogorov (1938), Yaglom (1947) and Sevastyanov (1951). They appear as a natural object when considering Markov processes that are certainly absorbed since they are invariant for the evolution of the distribution of the process conditioned on not being absorbed. They hence appropriately describe the state of the process at large times for non absorbed paths. Unlike invariant distributions for Markov processes, QSD are solutions of a non-linear equation and there can be 0, 1 or an infinity of them. Also, they cannot be obtained as Ces\`aro limits of Markovian dynamics. These facts make the computation of QSDs a nontrivial matter. We review different approximation methods for QSD that are useful for simulation purposes, mainly focused on Fleming-Viot dynamics. We also give some alternative proofs and extensions of known results.