Quasi-stationary distributions and population processes

Probability Surveys

Volumn 9, Issue 1, 2012, Pages 340-410

  Méléard, Sylvie ^a   Villemonais, Denis ^b  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b NONE

Author keywords

Birth and death process; Fleming Viot particle system.; Logistic feller diffusion; Population dynamics; Quasi stationarity; Ya glom limit

Indexed keywords

EID: 84877759353     PISSN: None     EISSN: 15495787     Source Type: Journal    
DOI: 10.1214/11-PS191     Document Type: Article

References (66)

1. Aldous, D., Stopping times and tightness. Ann. Probability, Volume 6, Issue 2 (1978), pages 335-340. MR0474446
2. Anderson, W. J., Continuous-time Markov chains. Springer-Verlag, New York, 1991. MR1118840
3. Asselah, A., Ferrari, P. A., and Groisman, P., Quasistationary distributions and Fleming-Viot processes in finite spaces. J. Appl. Probab., Volume 48, Issue 2 (2011), pages 322-332. MR2840302
4. Athreya, K. B. and Ney, P. E., Branching processes. Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 196. MR0373040
5. Bansaye, V., Surviving particles for subcritical branching processes in random environment. Stochastic Process. Appl., Volume 119, Issue 8 (2009), pages 2436-2464. MR2532207
6. Barbour, A. D. and Pollett, P. K., Total variation approximation for quasi-stationary distributions. J. Appl. Probab., Volume 47, Issue 4 (2010), pages 934-946. MR2752899
7. Berezin, F. A. and Shubin, M. A., The Schrödinger equation, volume 66 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1983 Russian edition by Yu. Rajabov, D. A. Le?ites and N. A. Sakharova and revised by Shubin, With contributions by G. L. Litvinov and Leǐites. MR1186643
8. Bieniek, M., Burdzy, K., and Finch, S., Non-extinction of a flemingviot particle model. Probab. Theory Related Fields (2011), pages 1-40. MR2925576
9. Billingsley, P., Convergence of probability measures. JohnWiley and Sons Inc., New York, 1968. MR0233396
10. Burdzy, K., Holyst, R., Ingerman, D., and March, P., Configurational transition in a fleming-viot-type model and probabilistic interpretation of laplacian eigenfunctions. J. Phys. A, Volume 29 (1996), pages 2633-2642.
11. Burdzy, K., Ho lyst, R., and March, P., A Fleming-Viot particle representation of the Dirichlet Laplacian. Comm. Math. Phys., Volume 214, Issue 3 (2000), pages 679-703. MR1800866
12. Carey, Liedo, P., Orozco, D., and Vaupel, J. W., Slowing of mortality rates at older ages in large medfly cohorts. Science, Volume 258 (1992), pages 457-461.
13. Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S., and San Martín, J., Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab., Volume 37, Issue 5 (2009), pages 1926-1969. MR2561437
14. Cattiaux P. and Méléard S., Competitive or weak cooperative stochastic lotka-volterra systems conditioned to non-extinction. J. Math. Biology, Volume 6 (2010), pages 797-829. MR2606515
15. Collet, P., Martínez, S., Mélíard, S., and San Martín, J., Quasistationary distributions for structured birth and death processes with mutations. Probab. Theory Related Fields, Volume 151, Issue 1 (2011), pages 191-231. MR2834717
16. Collet P. Martínez, S., and San Mart́n, J., Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption. Ann. Probab., Volume 23, Issue 3 (1995), pages 1300-1314. MR1349173
17. Coolen-Schrijner, P. and van Doorn, E. A., Quasi-stationary distributions for a class of discrete-time Markov chains. Methodol. Comput. Appl. Probab., Volume 8, Issue 4 (2006), pages 449-465. MR2329282
18. Darroch, J. N. and Seneta, E., On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Probability, Volume 2 (1965), pages 88-100. MR0179842
19. Darroch, J. N. and Seneta, E., On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probability, Volume 4 (1967), pages 192-196. MR0212866
20. Etheridge, A. M., Survival and extinction in a locally regulated population. Ann. Appl. Probab., Volume 14, Issue 1 (2004), pages 188-214. MR2023020
21. Ethier, S. N. and Kurtz, T. G., Markov processes, Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley and Sons Inc., New York, 1986. MR0838085
22. Ferrari, P. A., Kesten, H., Mart́nez, S., and Picco, P., Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab., Volume 23, Issue 2 (1995), pages 501-521. MR1334159
23. Ferrari, P. A. and Marić, N., Quasi stationary distributions and Fleming-Viot processes in countable spaces. Electron. J. Probab., Volume 12, Issue 24 (2007), pages 684-702. MR2318407
24. Ferrari, P. A., Martínez, S., and Picco, P., Some properties of quasi-stationary distributions in the birth and death chains: a dynamical approach. In Instabilities and nonequilibrium structures, III (Valparaíso, 1989), volume 64 of Math. Appl. (1991), pages 177-187. MR1177850
25. Fukushima, M., Dirichlet forms and Markov processes, volume 23 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1980. MR0569058
26. Galton, F. and Watson, H. W., On the probability of the extinction of families. Available at http://galton.org/essays/1870-1879/galton-1874-jaigi-family-extinction.p df, 1974.
27. Gantmacher, F. R., The theory of matrices. Vols. 1, 2. Translated by K. A. Hirsch. Chelsea Publishing Co., New York, 1959. MR0107649
28. Gompertz, B., On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Phil. Trans. R. Soc. London, Volume 115 (1825), pages 513-583.
29. Gong, G. L., Qian, M. P., and Zhao, Z. X., Killed diffusions and their conditioning. Probab. Theory Related Fields, Volume 80, Issue 1 (1988), pages 151-167. MR0970476
30. Good, P., The limiting behavior of transient birth and death processes conditioned on survival. J. Austral. Math. Soc., Volume 8 (1968), pages 716-722. MR0240879
31. Greenwood, M. and Irwin, J., The biostatistics of senility. Human Biology, Volume 11, Issue 1 (1939), pages 1-23.
32. Grigorescu, I. and Kang, M., Hydrodynamic limit for a Fleming-Viot type system. Stochastic Process. Appl., Volume 110, Issue 1 (2004), pages 111-143. MR2052139
33. Grigorescu, I. and Kang, M., Immortal particle for a catalytic branching process. Probab. Theory Related Fields (2011), pages 1-29. 10.1007/s00440-011-0347-6. MR2925577
34. Hart, A. G. and Pollett, P. K., New methods for determining quasistationary distributions for Markov chains. Math. Comput. Modelling, Volume 31, Issue 10-12 (2000), pages 143-150. MR1768777
35. Huillet, T., On Wright Fisher diffusion and its relatives. J. Stat. Mech.-Theory E., Volume 11 (2007), pages 6-+.
36. Ikeda, N. and Watanabe, S., Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, 1989. MR1011252
37. Joffe, A. and Métivier, M., Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. in Appl. Probab., Volume 18, Issue 1 (1986), pages 20-65. MR0827331
38. Karatzas I. and Shreve, S. E., Brownian motion and stochastic calculus, Volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1988. MR0917065
39. Karlin, S. and McGregor, J. L., The differential equations of birthand-death processes, and the Stieltjes moment problem. Trans. Amer.Math. Soc., Volume 85 (1957), pages 489-546. MR0091566
40. Knobloch, R. and Partzsch, L., Uniform conditional ergodicity and intrinsic ultracontractivity. Potential Analysis, Volume 33 (2010), pages 107-136. MR2658978
41. Kolb, M. and Steinsaltz, D., Quasilimiting behavior for onedimensional diffusions with killing. To appear in Ann. Probab. MR2917771
42. Lambert, A., The branching process with logistic growth. Ann. Appl. Probab., Volume 15, Issue 2 (2005), pages 1506-1535. MR2134113
43. Lambert, A., Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab., Volume 12, Issue 14 (2007), pages 420-446. MR2299923
44. Lladser, M. and San Martín, J., Domain of attraction of the quasistationary distributions for the Ornstein-Uhlenbeck process. J. Appl. Probab., Volume 37, Issue 2 (2000), pages 511-520. MR1781008
45. Mandl, P., Spectral theory of semi-groups connected with diffusion processes and its application. Czechoslovak Math. J., Volume 11, Issue 86 (1961), pages 558-569. MR0137143
46. Martínez, S., Picco, P., and San Martín, J., Domain of attraction of quasi-stationary distributions for the Brownian motion with drift. Adv. in Appl. Probab., Volume 30, Issue 2 (1998), pages 385-408. MR1642845
47. Martínez, S. and San Martín, J., Classification of killed onedimensional diffusions. Ann. Probab., Volume 32, Issue 1 (2004), pages 530-552. MR2040791
48. Pakes, A. G. and Pollett, P. K., The supercritical birth, death and catastrophe process: limit theorems on the set of extinction. Stochastic Process. Appl., Volume 32, Issue 1 (1989), pages 161-170. MR1008915
49. Pinsky, R. G., On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes. Ann. Probab., Volume 13, Issue 2 (1985), pages 363-378. MR0781410
50. Pollett, P., Quasi-stationary distributions: a bibliography. http://www.maths.uq.edu.au/~pkp/papers/qsds/qsds.pdf.
51. Pollett, P. K. and Stewart, D. E., An efficient procedure for computing quasi-stationary distributions of Markov chains with sparse transition structure. Adv. in Appl. Probab., Volume 26, Issue 1 (1994), pages 68-79. MR1260304
52. Renault, O., Ferriére, R., and Porter, J., The quasi-stationary route to extinction. Private communication.
53. Revuz, D. and Yor, M., Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999. MR1725357
54. Seneta, E. and Vere-Jones, D., On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probability, Volume 3 (1966), pages 403-434. MR0207047
55. Serre, D., Matrices, volume 216 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. Theory and applications, Translated from the 2001 French original. MR1923507
56. Steinsaltz, D. and Evans, S. N., Markov mortality models: Implications of quasistationarity and varying initial conditions. Theo. Pop. Bio., Volume 65 (2004), 319-337.
57. Steinsaltz, D. and Evans, S. N., Quasistationary distributions for onedimensional diffusions with killing. Trans. Amer. Math. Soc., Volume 359, Issue 3 (2007), pages 1285-1324. MR2262851
58. van Doorn, E. A., Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. in Appl. Probab., Volume 23, Issue 4 (1991), pages 683-700. MR1133722
59. van Doorn, E. A., Conditions for the existence of quasi-stationary distributions for birth-death processes with killing. Memorandum No. 1949, Department of Applied Mathematics, University of Twente, 2011.
60. van Doorn, E. A. and Pollett, P. K., Quasi-stationary distributions for reducible absorbing Markov chains in discrete time. Markov Process. Related Fields, Volume 15, Issue 2 (2009), pages 191-204. MR2538313
61. van Doorn, E. A. and Pollett, P. K., Quasi-stationary distributions. Memorandum No. 1945, Department of Applied Mathematics, University of Twente, 2011.
62. Vere-Jones, D., Some limit theorems for evanescent processes. Austral. J. Statist., Volume 11 (1969), pages 67-78. MR0263165
63. Verhulst, P. F., Notice sur la loi que la population suit dans son accroissement. Corr. Math. et Phys., 1938.
64. Villemonais, D., Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. Electronic Journal of Probability, Volume 16 (2011), pages 1663-1692. MR2835250
65. Villemonais, D. Interacting particle processes and approximation of Markov processes conditioned to not be killed. ArXiv e-prints, 2011. MR2835250
66. Yaglom, A. M., Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.), Volume 56 (1947), pages 795-798. MR0022045