The Annals of Probability

Quasilimiting behavior for one-dimensional diffusions with killing

Martin Kolb and David Steinsaltz

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Abstract

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.

Article information

Source
Ann. Probab. Volume 40, Number 1 (2012), 162-212.

Dates
First available in Project Euclid: 3 January 2012

Permanent link to this document
http://projecteuclid.org/euclid.aop/1325605001

Digital Object Identifier
doi:10.1214/10-AOP623

Mathematical Reviews number (MathSciNet)
MR2917771

Zentralblatt MATH identifier
1278.60121

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47) 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

Keywords
Killed one-dimensional diffusions quasi-limiting distributions

Citation

Kolb, Martin; Steinsaltz, David. Quasilimiting behavior for one-dimensional diffusions with killing. Ann. Probab. 40 (2012), no. 1, 162--212. doi:10.1214/10-AOP623. http://projecteuclid.org/euclid.aop/1325605001.


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