Simulation of quasi-stationary distributions on countable spaces

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 Cesàro 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.

References in corpus (7)

• Shift in the velocity of a front due to a cut-off
• Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization
• The genealogy of branching Brownian motion with absorption
• Noisy traveling waves: effect of selection on genealogies
• Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions
• Quasi-stationary simulation of the contact process
• Branching Brownian motion with selection of the N right-most particles: An approximate model

Cited by in corpus (3)

• Central Limit Theorem for stationary Fleming--Viot particle systems in finite spaces
• Front propagation and quasi-stationary distributions: the same selection principle?
• Uniform in time propagation of chaos for a Moran model