Open Access
January 2012 Quasilimiting behavior for one-dimensional diffusions with killing
Martin Kolb, David Steinsaltz
Ann. Probab. 40(1): 162-212 (January 2012). 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.


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Martin Kolb. David Steinsaltz. "Quasilimiting behavior for one-dimensional diffusions with killing." Ann. Probab. 40 (1) 162 - 212, January 2012.


Published: January 2012
First available in Project Euclid: 3 January 2012

zbMATH: 1278.60121
MathSciNet: MR2917771
Digital Object Identifier: 10.1214/10-AOP623

Primary: 60J60 , 60J70
Secondary: 47E05 , 47F05 , 60J35

Keywords: Killed one-dimensional diffusions , quasi-limiting distributions

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 1 • January 2012
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