Quasi-Stationary Distributions of Interacting Dynamical Systems and their approximation (2506.21087v1)

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.