Quasi-Stationary Distributions of Interacting Dynamical Systems and their approximation

TL;DR

This paper extends algorithms for approximating Quasi-Stationary Distributions to McKean-Vlasov dynamics, providing theoretical guarantees, tightness results, and simulations for non-linear interacting systems.

Contribution

It introduces conditions for QSD approximation in McKean-Vlasov dynamics and analyzes their behavior in both compact and non-compact cases.

Findings

Established conditions ensuring weak limits are QSDs

Proved tightness results for non-compact cases

Analyzed behavior of QSDs as step size approaches zero

Abstract

In \cite{BCP}, the authors built and studied an algorithm based on the (self)-interaction of a dynamics with its occupation measure to approximate Quasi-Stationary Distributions (QSD) of general Markov chains conditioned to stay in a compact set. In this paper, we propose to tackle the case of McKean-Vlasov-type dynamics, \emph{i.e.} of dynamics interacting with their marginal distribution (conditioned to not be killed). In this non-linear setting, we are able to exhibit some conditions which guarantee that weak limits of these sequences of random measures are QSDs of the given dynamics. We also prove tightness results in the non-compact case. These general conditions are then applied to Euler schemes of McKean-Vlasov SDEs and in the compact case, the behavior of these QSDs when the step $h$ goes to $0$ is investigated. Our results also allow to consider some examples in the non-compact case and some new tightness criterions are also provided in this setting. Finally, we illustrate our theoretical results with several simulations.