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Quasistationary distributions for one-dimensional diffusions with killing

Authors: David Steinsaltz and Steven N. Evans
Journal: Trans. Amer. Math. Soc. 359 (2007), 1285-1324
MSC (2000): Primary 60J60
Published electronically: October 24, 2006
MathSciNet review: 2262851
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Abstract: 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 $ \infty$. 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.

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Additional Information

David Steinsaltz
Affiliation: Department of Demography, University of California, 2232 Piedmont Ave., Berkeley, California 94720

Steven N. Evans
Affiliation: Department of Statistics #3860, Evans Hall 367, University of California, Berkeley, California 94720-3860

Received by editor(s): May 19, 2004
Received by editor(s) in revised form: December 26, 2004
Published electronically: October 24, 2006
Additional Notes: The first author was supported by Grant K12-AG00981 from the National Institute on Aging. The second author was supported in part by Grants DMS-00-71468 and DMS-04-05778 from the National Science Foundation, and by the Miller Institute for Basic Research in Science.
Article copyright: © Copyright 2006 David Steinsaltz and Steven N. Evans

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