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Persistence of a critical super-2 process
Published online by Cambridge University Press: 14 July 2016
Abstract
It is shown that the critical two-level (2, d, 1, 1)-superprocess is persistent in dimensions d greater than 4. This complements the extinction result of Wu (1994) and implies that the critical dimension is 4.
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- Copyright © Applied Probability Trust 1995
References
[2]
Dawson, D. A. (1992) Infinitely divisible random measures and superprocesses. In Stochastic Analysis and Related Topics, Progr. Prob.
32, pp. 1–129. Birkhäuser, Basel.Google Scholar
[3]
Dawson, D. A. (1993) Measure-valued Markov processes. In Ecole d'Eté de Probabilités de Saint Flour XXI-1991, pp. 1–260. Lecture Notes in Mathematics 1541, Springer-Verlag, Berlin.Google Scholar
[4]
Dawson, D. A. and Hochberg, K. J. (1991) A multilevel branching model. Adv. Appl. Prob.
23, 701–715.CrossRefGoogle Scholar
[5]
Dawson, D. A., Hochberg, K. J. and Vinogradov, V. (1994) On path properties of super-2 processes I. In Measure-Valued Processes, Stochastic Partial Differential Equations and Interacting Systems.
CRM Proceedings and Lecture Notes 5, pp. 69–82. American Mathematical Society, Providence, RI.Google Scholar
[6]
Dawson, D. A., Hochberg, K. J. and Vinogradov, V. (1994) On path properties of super-2 processes II. In Proceedings of Symposia in Pure Mathematics
57, pp. 385–403. American Mathematical Society, Providence, RI.Google Scholar
[7]
Dawson, D. A., Hochberg, K. J. and Wu, Y. (1990) Multilevel branching systems. In White Noise Analysis: Mathematics and Applications, pp. 93–107. World Scientific, Singapore.Google Scholar
[8]
Dawson, D. A. and Perkins, E. A. (1991) Historical Processes.
Memoirs of the American Mathematical Society 454, Providence, RI.Google Scholar
[9]
Dynkin, E. B. (1989) Three classes of infinite dimensional diffusions. J. Functional Anal.
56, 75–110.CrossRefGoogle Scholar
[10]
Gorostiza, L. G. and Wakolbinger, A. (1991) Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Prob.
19, 266–288.Google Scholar
[11]
Gorostiza, L. G. and Wakolbinger, A. (1994) Long time behavior of critical branching particle systems and applications. In Measure-Valued Processes, Stochastic Partial Differential Equations and Interacting Systems.
CRM Proceedings and Lecture Notes 5, pp. 119–137. American Mathematical Society, Providence, RI.Google Scholar
[12]
Gorostiza, L. G., Roelly, S. and Wakolbinger, A. (1990) Sur la persistence du processes de Dawson-Watanabe stable. L'interversion de la limite en temps et de la renormalisation. In Séminaire de Probabilités
XXIV, pp. 275–281. Lecture Notes in Mathematics 1426, Springer-Verlag, Berlin.Google Scholar
[13]
Hochberg, K. J. (1994) Hierarchically structured branching populations with spatial motion. Rocky Mountain J. Math., To appear.Google Scholar
[14]
Kallenberg, O. (1983) Random Measures, 3rd edn.
Akademie-Verlag, Berlin; Academic Press, New York.Google Scholar
[15]
Wu, Y. (1992) Dynamic Particle Systems and Multilevel Measure Branching Processes. , Carleton University.Google Scholar
[16]
Wu, Y. (1993) A multilevel birth-death particle system and its continuous diffusion. Adv. Appl. Prob.
25, 549–569.Google Scholar
[17]
Wu, Y. (1994) Asymptotic behaviour of the two level measure branching process. Ann. Prob.
22, 854–874.Google Scholar
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