Quasistationary distributions for one-dimensional diffusions with killing

Transactions of the American Mathematical Society

Volumn 359, Issue 3, 2007, Pages 1285-1324

  Steinsaltz, David ^a,b   Evans, Steven N ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b QUEEN S UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 38949165926     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-06-03980-8     Document Type: Article

References (41)

1. James J. Anderson, A vitality-based model relating stressors and environmental properties to organism survival, Ecological Monographs 70 (2000), no. 3, 445-70.
2. Milton Abramowitz and Irene Stegun, Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover, New York, 1965. MR0208797 (34:8606)
3. Andrei N. Borodin and Paavo Salminen, Handbook of Brownian motion-facts and formulae,Birkhauser, Basel, 1996. MR1477407 (98i:60077)
4. James A. Cavender, Quasi-stationary distributions of birth-and-death processes,Ad-vances in Applied Probability 10 (1978), no. 3, 570-586. MR0501388 (58:18754)
5. Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York, Toronto, London, 1955. MR0069338 (16:1022b)
6. James R. Carey, Pablo Liedo, Dina Orozco, and James W. Vaupel, Slowing of mortality rates at older ages in large medfly cohorts, Science 258 (1992), no. 5081, 457-461. (Pubitemid 23397778)
7. Pierre Collet, Servet Martinez, and Jaime San Martin, Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption, The Annals of Probability 23 (1995), no. 3, 1300-1314. MR1349173 (96i:60083)
8. Erik A. Van Doorn, Quasistationary distributions and convergence to quasi-stationarity ofbirth-death processes, Advances in Applied Probability 23 (1991), 683-700. MR1133722 (93b:60192)
9. J. N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing discrete-time finite Markov chains, Journal of Applied Probability 2 (1965), 88-100. MR0179842 (31:4083)
10. R. E. Edwards, Functional analysis, Dover, New York, 1995, Theory and applications, Corrected reprint of the 1965 original. MR1320261 (95k:46001)
11. Stewart Ethier and Thomas Kurtz, Markov processes: Characterization and convergence, John Wiley & Sons, 1986. MR0838085 (88a:60130)
12. William Feller, The parabolic differential equations and the associated semi-groups of transformations, Annals of Mathematics 55 (1952), 468-519. MR0047886 (13:948a)
13. P. A. Ferrari, H. Kesten, S. Martinez, and P. Picco, Existence of quasi-stationary distributions. A renewal dynamical approach, The Annals of Probability 23 (1995), no. 2, 501-521. MR1334159 (96c:60089)
14. Leonid A. Gavrilov and Natalia S. Gavrilova, The biology of lifespan: A quantitative approach, Harwood Academic Publishers, Chur, Switzerland, 1991.
15. Phillip Good, The limiting behavior of transient birth and death processes conditioned on survival, Journal of the Australian Mathematical Society 8 (1968), 716-722. MR0240879 (39:2224)
16. Frederic Gosselin, Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology, Annals of Applied Probability 11 (2001), no. 1, 261-284. MR1825466 (2002b:60122) (Pubitemid 33280273)
17. Goran Hognas, On the quasi-stationary distribution of a stochastic Ricker model, Stochastic Processes and their Applications 70 (1997), no. 2, 243-263. MR1475665 (98k:60150) (Pubitemid 127161562)
18. Kiyosi Ito and Henry P. McKean, Jr., Diffusion processes and their sample paths, Springer-Verlag, New York, Heidelberg, Berlin, 1965. MR0199891 (33:8031)
19. J. F. C. Kingman, The exponential decay of Markov transition probabilities, Proceedings of the London Mathematical Society. Third Series 13 (1963), 337-358. MR0152014 (27:1995)
20. Masaaki Kijima and E. Seneta, Some results for quasi-stationary distributions of birth-death processes, Journal of Applied Probability 28 (1991), no. 3, 503-511. MR1123824 (92j:60103)
21. H. Le Bras, Lois de mortaliteet age limite, Population 33 (1976), no. 3, 655-691.
22. Jean B. Lasserre and Charles E. M. Pearce, On the existence of a quasistationary measure for a Markov chain, The Annals of Probability 29 (2001), no. 1, 437-446. MR1825158 (2002c:60114) (Pubitemid 33279255)
23. Petr Mandl, Spectral theory of semi-groups connected with diffusion processes and its application, Czechoslovak Mathematical Journal 11 (1961), no. 86, 558-569. MR0137143 (25:599)
24. Servet Martinez andJaime SanMartin, Rates of decay and h-processes for one dimensional diffusions conditioned on non-absorption, Journal of Theoretical Probability 14 (2001), no. 1, 199-212. MR1822901 (2002b:60134) (Pubitemid 33263500)
25. -, Classification of killed one-dimensional diffusions, The Annals of Probability 32 (2004), no. 1, 530-552. MR2040791 (2005e:60180)
26. Servet Martinez, Pierre Picco, and Jaime San Martin, Domain of attraction of quasi-stationary distributions for the Brownian motion with drift, Advances in Applied Probability 30 (1998), no. 2, 385-408. MR1642845 (99i:60156) (Pubitemid 128482489)
27. Roger Nussbaum, Positive operators and elliptic eigenvalue problems, Mathematische Zeitschrift 186 (1984), 247-264. MR0741305 (85j:35136)
28. Ross G. Pinsky, On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes,The Annals of Probability 13 (1985), no. 2, 363-378. MR0781410 (86i:60201)
29. -, Positive harmonic functions and diffusion, Cambridge University Press, Cambridge, 1995. MR1326606 (96m:60179)
30. S. T. Rachev, Probability metrics, Wiley, 1992. MR1105086 (93b:60012)
31. Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York, 1972. MR0493419 (58:12429a)
32. David Steinsaltz and Steven N. Evans, Markov mortality models: Implications of quasistationarity and varying initial conditions, Theoretical Population Biology 65 (2004), no. 4, 319-337. (Pubitemid 38611002)
33. E. Seneta, Quasi-stationary distributions and time-reversion in genetics. (With discussion), Journal of the Royal Statistical Society Series B 28 (1966), 253-277. MR0202192 (34:2065)
34. E. Seneta, Non-negative matrices, Wiley, New York, 1973. MR0389944 (52:10773)
35. E. Seneta and D. Vere-Jones, On quasi-stationary distributions in discrete time for Markov chains with a denumerable infinity of states, Journal of Applied Probability 3 (1966), 403-434. MR0207047 (34:6863)
36. Richard Tweedie, r-theory for Markov chains on a general state space I: Solidarity properties and r-recurrent chains, The Annals of Probability 2 (1974), 840-864. MR0368151 (51:4393)
37. James W. Vaupel, Trajectories of mortality at advanced ages, Between Zeus and the Salmon: The Biodemography of Longevity (Kenneth W. Wachter and Caleb E. Finch, eds.), National Academies Press, Washington, D.C., 1997, pp. 17-37.
38. D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quarterly Journal of Mathematics. Oxford. Second Series 13 (1962), 7-28. MR0141160 (25:4571)
39. Joshua Weitz and Hunter Fraser, Explaining mortality rate plateaus, Proc. Natl. Acad. Sci. USA 98 ( 2001), no. 26, 15383-15386. (Pubitemid 34014023)
40. A. M. Yaglom, Certain limit theorems of the theory of branching random processes, Doklady Akad. Nauk SSSR (N.S.) 56 (1947), 795-798. MR0022045 (9:149e)
41. Kosaku Yosida, Lectures on differential and integral equations, Dover, New York, 1991. MR1105769 (92a:34002)