Quasi-Stationary Distributions in Stochastic Systems

The notion of quasi-stationarity arises in stochastic processes that admit at least one absorbing state. A quasi-stationary distribution (QSD) characterises the conditional law of the system given non-absorption over long time horizons, capturing a metastable regime preceding eventual absorption. Unlike classical stationary distributions, which describe equilibrium in recurrent systems, QSDs describe apparent equilibrium in transient systems where absorption (extinction or failure) is certain but may be delayed. Existence and uniqueness of QSDs hinge on spectral properties of the generator of the underlying process, often requiring compactness or positivity conditions to secure a spectral gap. Convergence to a QSD may be exponential or polynomial, depending on drift, dimensionality and boundary behaviour. Computation of QSDs has driven the development of particle algorithms and stochastic approximation schemes, linking theoretical advances to applications in ecology (population extinction risk), epidemiology (persistence of disease), chemical kinetics (metastable states in reaction networks) and Bayesian computation (sampling from posterior distributions via killed diffusions). Recent work has emphasised non-linear Lyapunov criteria and Monte Carlo methods to obtain quantitative convergence rates, while advances in spectral theory underpin new insights into uniform convergence across initial conditions. These developments have deepened our understanding of longevity and extinction phenomena in diverse stochastic systems.

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Technical terms

Quasi-Stationary Distribution: A conditional probability law over transient states that remains invariant under the evolution of a process conditioned on non-absorption.

Absorbing State: A state of a stochastic process from which no exit is possible, representing extinction or absorption events.

Yaglom Limit: The limiting conditional distribution of a process given survival up to a large time horizon, often coinciding with the QSD.

Fleming-Viot Process: An interacting particle system that approximates a QSD by redistributing mass from absorbed particles to survivors, preserving the number of active particles.

References

1. Uniform convergence to the $Q$-process. Electronic Communications in Probability (2017).
2. Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes. Stochastic Processes and their Applications (2021).
3. A stochastic approximation approach to quasi-stationary distributions on finite spaces. Electronic Communications in Probability (2015).
4. An approximation scheme for quasi-stationary distributions of killed diffusions. Stochastic Processes and their Applications (2020).
5. General criteria for the study of quasi-stationarity. Electronic Journal of Probability (2023).
6. Quasi Stationary Distributions and Fleming-Viot Processes in Countable Spaces. Electronic Journal of Probability (2007).
7. Interacting Particle Systems and Yaglom Limit Approximation of Diffusions with Unbounded Drift. Electronic Journal of Probability (2011).
8. Polynomial rate of convergence to the Yaglom limit for Brownian motion with drift. Electronic Communications in Probability (2020).
9. Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: Uniform estimates in a compact soft case. ESAIM Probability and Statistics (2022).