QUASILIMITING BEHAVIOR FOR ONE-DIMENSIONAL DIFFUSIONS WITH KILLING

• 成果类型: Article
• 署名作者: Kolb, Martin; Steinsaltz, David
• 署名单位: University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP623
• 发表日期: 2012
• 页码: 162-212
• 关键词: quasi-stationary distributions; markov-chains; heat kernel; large time; EXISTENCE; Operators; survival; THEOREM; models

摘要

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