Quasistationary distributions for one-dimensional diffusions with killing

By David Steinsaltz, Steven N. Evans. 2004

67 citations recorded by the source corpus.

We extend some results on the convergence of one-dimensional diffusions killed at the boundary, conditioned on extended survival, to the case of general killing on the interior. We show, under fairly general conditions, that a diffusion conditioned on long survival either runs off to infinity almost surely, or almost surely converges to a quasistationary distribution given by the lowest eigenfunction of the generator. In the absence of internal killing, only a sufficiently strong inward drift can keep the process close to the origin, to allow convergence in distribution. An alternative, that arises when general killing is allowed, is that the conditioned process is held near the origin by a high rate of killing near oo. We also extend, to the case of general killing, the standard result on convergence to a quasistationary distribution of a diffusion on a compact interval.

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

View Paper: https://www.semanticscholar.org/paper/a01df6fc074c3b271b0ef67ad2f700a1c34a8a99

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Citations in the corpus

Quasi-stationary distributions and population processes (2011, methodology): “For recent development of the theory of QSDs for diffusion processes, we refer to Steinsaltz and Evans [57] and Kolb and Steinsaltz [41] where the case of one ha l-0 06 53 83 4, v er si on 1 - 20 D ec 2 01 1 dimensional diffusions with different boundary conditions is studied.”
Quasi-stationary distributions and diffusion models in population dynamics (2007, methodology): “For Kolmogorov diffusions, the theory started with Mandl’s paper [22] in 1961, and was then developed by many authors (see, in particular, [3, 23, 30]).”
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].”