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Quasilimiting behavior for one-dimensional diffusions with killing

Kolb, M. and Steinsaltz, D. (2012) Quasilimiting behavior for one-dimensional diffusions with killing. Annals of Probability, 40 (1). pp. 162-212. ISSN 0091-1798

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To link to this item DOI: 10.1214/10-AOP623


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 ∞ is larger than the principal eigenvalue, then the eigenfunction is always integrable. When the killing at ∞ 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.

Item Type:Article
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:29094
Publisher:Institute of Mathematical Statistics

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