Branching processes with immigration and related topics

Frontiers of Mathematics in China

Volumn 1, Issue 1, 2006, Pages 73-97

  Li, Zeng Hu ^a  

^a BEIJING NORMAL UNIVERSITY   (China)

Author keywords

Affine process; Branching process; Fluctuation limit; Immigration; Measure valued process; Ornstein Uhlenbeck process; Skew convolution semigroup; Stochastic equation

Indexed keywords

EID: 33746137193     PISSN: 16733452     EISSN: 16733576     Source Type: Journal    
DOI: 10.1007/s11464-005-0027-x     Document Type: Review

References (78)

1. Athreya K. B. and Ney P. E., Branching Processes, Berlin Heidelberg New York: Springer, 1972
2. Barros-Neto J., An Introduction to the Theory of Distributions, New York: Marcel Dekker, 1973 (Chinese translation, 1981)
3. Bertoin J., Lévy Processes, Cambridge: Cambridge University Press, 1996
4. Bogachev V. I., Röckner M. and Schmuland B., Generalized Mehler semigroups and applications, Probab. Theory Relat. Fields, 1996, 105: 193-225
5. Bojdecki T. and Gorostiza L. G., Langevin equation for L1-valued Gaussian processes and fluctuation limits of infinite particle systems, Probab. Theory Relat. Fields, 1986, 73: 227-244
6. Chen M.-F., Ergodic convergence rates of Markov processes - eigenvalues, inequalities and ergodic theory. In: Proceedings of ICM 2002, Vol. III, Beijing: Higher Education Press, 2002, 25-40
7. Chen M.-F., From Markov Chains to Non-Equilibrium Particle Systems, 2nd edn., Singapore: World Scientific, 2004
8. Chen, M.-F., Eigenvalues, Inequalities and Ergodic Theory, London: Springer, 2004
9. Dawson D. A., Measure-valued Markov Processes. In: Mannequin P. L. (ed.), Lecture Notes in Math., Vol. 1541, Berlin Heidelberg New York: Springer, 1993, 1-260
10. Dawson D. A. and Fleischmann K., Critical branching in a highly fluctuating random medium, Probab. Theory Relat. Fields, 1991, 90: 241-274
11. Dawson D. A. and Fleischmann K., A continuous super-Brownian motion in a super-Brownian medium, J. Theor. Probab., 1997, 10: 213-276
12. Dawson D. A., Fleischmann K. and Gorostiza L. G., Stable hydrodynamic limit fluctuations of a critical branching particle system in a random medium, Ann. Probab., 1989, 17(3), 1083-1117
13. Dawson D. A. and Li Z. H., Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions, Probab. Theory Relat. Fields, 2003, 127: 37-61
14. Dawson D. A. and Li Z. H., Non-differentiable skew convolution semigroups and related Ornstein-Uhlenbeck processes, Potential Anal., 2004, 20: 285-302
15. Dawson D. A. and Li Z. H., Skew convolution semigroups and affine Markov processes, Ann. Probab., 2006 (in press)
16. Dawson D. A., Li Z. H., Schmuland B., and Sun W., Generalized Mehler semigroups and catalytic branching processes with immigration, Potential Anal., 2004, 21: 75-97
17. Dawson D. A., Li Z. H. and Wang H., Superprocesses with dependent spatial motion and general branching densities, Electron. J. Probab., 2001, 6(25), 1-33
18. Dellacherie C. and Meyer P. A., Probabilities and Potential, Chap. V-VIII, Amsterdam: North-Holland, 1982
19. Duffie D., Filipović D., and Schachermayer W., Affine processes and applications in finance, Ann. Appl. Probab., 2003, 13: 984-1053
20. Dynkin E. B., Diffusions, Superdiffusions and Partial Differential Equations, Providence, RI: Amer. Math. Soc., 2002
21. Etheridge A. M., An Introduction to Superprocesses, Providence, RI: Amer. Math. Soc., 2000
22. Fitzsimmons P. J., Construction and regularity of measure-valued Markov branching processes, Isr. J. Math., 1988, 64:337-361
23. Fitzsimmons P. J., On the martingale problem for measure-valued Markov branching processes. In: Seminar on Stochastic Processes, Vol. 1991, Boston, MA.: Birkhäuser Boston, 1992, 39-51
24. Fu Z. F. and Li Z. H., Measure-valued diffusions and stochastic equations with Poisson process, Osaka J. Math., 2004, 41: 727-744
25. Fuhrman M. and Röckner M., Generalized Mehler semigroups: the non-Gaussian case, Potential Anal., 2000, 12: 1-47
26. Gorostiza L. G. and Li Z. H., Fluctuation limits of measure-valued immigration processes with small branching. In: Gonzalez-Burrios J. M. and Gorostiza L. G. (eds.), Aportaciones Matemáticas: Investigación, Vol. 14, Mexico: Sociedad Matemática Mexicana, 261-268
27. Gorostiza L. G. and Li Z. H., High density fluctuations of immigration branching particle systems. In: CMS Conference Proceedings, Ser. 26, Providence, RI: Amer. Math. Soc., 2000, 159-171
28. Grey D. R., Asymptotic behavior of continuous time, continuous state-space branching processes, J. Appl. Probab., 1974, 11: 669-677
29. Harris T. E., The Theory of Branching Processes, Berlin Heidelberg New York: Springer, 1963
30. Hong W. M., Long time behavior for the occupation time processes of a super-Brownian motion with random immigration, Stoch. Process. Appl., 2002, 102: 43-62
31. Hong W. M., Moderate deviation for the super-Brownian motion with super-Brownian immigration, J. Appl. Probab., 2002, 39: 829-838
32. Hong W. M., Large deviations for the super-Brownian motion with super-Brownian immigration, J. Theor. Probab., 2003, 16: 899-922
33. Hong W. M., Quenched mean limit theorems for the super-Brownian motion with super-Brownian immigration, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2005, 8: 383-396
34. Hong W. M. and Li Z. H., A central limit theorem for super Brownian motion with super Brownian immigration, J. Appl. Probab., 1999, 36: 1218-1224
35. Hong W. M. and Li Z. H., Large and moderate deviations for occupation times of immigration superprocesses, 2005, 8: 593-603
36. Ikeda N. and Watanabe S., Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland, 1989
37. Jiřina M., Stochastic branching processes with continuous state space, Czechoslov. Math. J., 1958, 8: 292-313
38. Jiřtna M., Branching processes with measure-valued states. In: Trans. Third Prague Conf. Information Theory, Statist. Decision Func., Random Process., Prague: Publ. House Czech. Acad. Sci., 1964, 333-357
39. Kawazu K. and Watanabe S., Branching processes with immigration and related limit theorems, Theory Probab. Appl., 1971, 16: 36-54
40. Le Gall J.-F., Spatial Branching Processes, Random Snakes and Partial Differential Equations, Basel: Birkhäuser, 1999 (Lectures in Mathematics ETH Zürich)
41. Le Gall J. -F. and Le Jan Y., Branching processes in Lévy processes: the exploration process, Ann. Probab., 1998, 26: 213-252
42. Li Z. H., Integral representations of continuous functions. Chinese Sci. Bull. (Chinese ed.), 1991. 36: 81-84, math.bnu.edu.cn/~lizh
43. Li Z. H., Integral representations of continuous functions. Chinese Sci. Bull. (English edn.: 1991, 36: 979-983)
44. Li Z. H., Measure-valued branching processes with immigration, Stoch. Process. Appl., 1992, 43: 249-264
45. Li Z. H., Convolution semigroups associated with measure-valued branching processes, Chin. Sci. Bull. (Chinese edn.), 1995, 40: 2018-2021, math.bnu.edu.cn/~lizh
46. Li Z. H., Convolution semigroups associated with measure-valued branching processes, Chin. Sci. Bull. (English edn.: 1996, 41: 276-280)
47. Li Z. H., Immigration structures associated with Dawson-Watanabe superprocesses, Stoch. Process. Appl., 1996, 62: 73-86
48. Li Z. H., Immigration processes associated with branching particle systems, Adv. Appl. Probab., 1998, 30: 657-675
49. Li Z. H., Measure-valued immigration diffusions and generalized Ornstein-Uhlenbeck diffusions, Acta Math. Appl. Sin., 1999, 15: 310-320
50. Li Z. H., Asymptotic behavior of continuous time and state branching processes, J. Aust. Math. Soc., Ser. A, 2000, 68: 68-84
51. Li Z. H., Ornstein-Uhlenbeck type processes and branching processes with immigration, J. Appl. Probab., 2000, 37: 627-634
52. Li Z. H., Skew convolution semigroups and related immigration processes, Theory Probab. Appl., 2002, 46: 274-296
53. Li Z. H., A limit theorem of discrete Galton-Watson branching processes with immigration, J. Appl. Probab., 2006 (in press)
54. Li Z. H. and Shiga T., Measure-valued branching diffusions: immigrations, excursions and limit theorems, J. Math. Kyoto Univ., 1995, 35: 233-274
55. Li Z. H. and Wang Z. K., Measure-valued branching process and immigration processes, Adv. Math. (China), 1999, 28: 105-134
56. Li Z. H. and Zhang M., Fluctuation limit theorems of immigration superprocesses with small branching, Stat. Probab. Lett., 2006 (in press)
57. Linde W., Probability in Banach Spaces - Stable and Infinitely Divisible Distributions, New York: Wiley, 1986
58. van Neerven J. M. A. M., Continuity and representation of Gaussian Mehler semigroups, Potential Anal., 2000, 13: 199-211
59. Pakes A. G., Some limit theorems for continuous-state branching processes, J. Aust. Math. Soc., Ser. A, 1988, 44: 71-87
60. Pakes A. G., Revisiting conditional limit theorems for mortal simple branching processes, Bernoulli, 1999, 5: 969-998
61. Pitman J. and Yor M., A decomposition of Bessel bridges, Z. Wahrscheinlichkeitstheor. Verw. Geb., 1982, 59: 425-457
62. Röckner M. and Wang F. Y., Harnack and functional inequalities for generalized Mehler semigroups, J. Funct. Anal., 2003, 203: 237-261
63. Sato K., Lévy Processes and Infinitely Divisible Distributions, Cambridge: Cambridge University Press, 1999
64. t Stat. Probab. Lett., 2001, 52: 183-188
65. Schaefer H. H., Topological Vector Spaces, Berlin Heidelberg New York: Springer, 1980
66. Sharpe M. J., General Theory of Markov Processes, New York: Academic Press, 1988
67. Shiga T. and Watanabe S., Bessel diffusions as a one-parameter family of diffusion processes, Z. Wahrscheinlichkeitstheor. Verw. Geb., 1973, 27: 37-46
68. Walsh J. B., An Introduction to Stochastic Partial Differential Equations. In: Lecture Notes in Math., Vol. 1180, Berlin Heidelberg New York: Springer, 1986, 226-439
69. Wang F. Y., Functional Inequalities, Markov Processes, and Spectral Theory, Beijing: Science Press, 2005
70. Wang F. Y., The stochastic order and critical phenomena for surperprocesses, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2005 (in press)
71. Wang F. Y., Dimension-free Harnack inequalities and applications, Front. Math. China, 2006, 1
72. Wang H., State classification for a class of measure-valued branching diffusions in a Brownian medium, Probab. Theory Relat. Fields, 1997, 109: 39-55
73. Watanabe S., A limit theorem of branching processes and continuous state branching processes, J. Math. Kyoto Univ., 1968, 141-167
74. Zhang M., Large deviation for super-Brownian motion with immigration, J. Appl. Probab., 2004, 41: 187-201
75. Zhang M., Moderate deviations for super-Brownian motion with immigration, Sci. China, Ser. A, 2004, 41:440-452
76. Zhang M., Moderate deviation principles for the occupation time process of a super-Brownian motion with immigration, Chin. J. Contemp. Math., 2005, 26: 61-70
77. Zhang M., On the large deviation for Brownian branching particle system, J. Appl. Probab., 2005 (in press)
78. (with the announcement to appear in Front. Math. China, 2006, 1)