Quasilimiting behavior for one-dimensional diffusions with killing

By Martin Kolb, David Steinsaltz. 2010

48 citations recorded by the source corpus.

This paper extends and clarifies results of Steinsaltz and Evans [Trans. Amer. Math. Soc. 359 (2007) 1285-1234], which found conditions for convergence of a killed one-dimensional diffusion conditioned on survival, to a quasistationary distribution whose density is given by the principal eigenfunction of the generator. Under the assumption that the limit of the killing at infinity differs from the principal eigenvalue we prove that convergence to quasistationarity occurs if and only if the principal eigenfunction is integrable. When the killing at infinity is larger than the principal eigenvalue, then the eigenfunction is always integrable. When the killing at infinity is smaller, the eigenfunction is integrable only when the unkilled process is recurrent; otherwise, the process conditioned on survival converges to 0 density on any bounded interval.

Subfields: Mathematical Physics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/9e7e5c98fe8cef6e41d5532029fe627057544a4b

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Citations in the corpus

Quasi-stationary distributions and population processes (2011, background): “Keywords: population dynamics, quasi-stationarity, Yaglom limit, birth and death process, logistic Feller diffusion, Fleming-Viot particle system.”
General criteria for the study of quasi-stationarity (2017, methodology): “One may for instance use the bounds for the first eigenvalue of the (Dirichlet) infinitesimal generator of (X t , t ≥ 0) obtained in a L2 (symmetric) setting using Rayleigh-Ritz formula in [74, 86, 87], as observed in [58].”
Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes (2017, background): “The quasistationary behavior of (extensions of) these two models have received a lot of attention in the one-dimensional case [34, 33, 25, 29, 3, 14, 15, 22].”