quasistationary

Quasistationary describes a regime or distribution in stochastic systems that persistently resembles a stationary state when conditioned on not having exited a given set, often an absorbing state. In Markov processes with absorbing states, the system may quickly drift toward extinction or absorption, but after that initial period, the distribution over remaining (transient) states can settle into a pattern that appears time-invariant if one only looks at trajectories that have not yet been absorbed.

Formally, for a Markov process with absorbing state(s) and τ denoting the absorption time, a quasi-stationary distribution (QSD) μ is a probability distribution on the transient states such that if X0 is drawn from μ, then for all t ≥ 0 the conditional distribution of Xt given τ > t equals μ. In continuous time, μ is typically an eigenvector of the subgenerator corresponding to the principal eigenvalue, reflecting an exponential decay of the non-absorption probability with rate λ.

Quasistationarity is distinct from true stationarity. A stationary distribution for the full chain remains invariant under time evolution without regard to absorption, whereas a QSD is not stationary for the whole process because mass eventually leaks into the absorbing state. It is, however, stationary conditional on survival, hence the term quasistationary.

Applications and examples include population dynamics where extinction is possible, chemical reaction networks with metastable intermediates, and certain epidemiological models where disease fade-out is eventual but the transient dynamics are long-lived. In physics, quasi-stationary states describe metastable systems that persist for long times before transitioning to a different state. Computational methods for QSDs include spectral analysis of transition operators, fixed-point iterations, and stochastic simulations that condition on non-absorption.