Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-24T01:52:08.710Z Has data issue: false hasContentIssue false
  • English
  • Français

Immigration processes associated with branching particle systems

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Zeng-Hu Li
Zeng-Hu Li*
Affiliation:
Beijing Normal University
*
∗ Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China. Email address: lizh@email.bnu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

The immigration processes associated with a given branching particle system are formulated by skew convolution semigroups. It is shown that every skew convolution semigroup corresponds uniquely to a locally integrable entrance law for the branching particle system. The immigration particle system may be constructed using a Poisson random measure based on a Markovian measure determined by the entrance law. In the special case where the underlying process is a minimal Brownian motion in a bounded domain, a general representation is given for locally integrable entrance laws for the branching particle system. The convergence of immigration particle systems to measure-valued immigration processes is also studied.

Keywords

Branching particle systemimmigrationDawson-Watanabe superprocessskew convolution semigroupentrance lawminimal Brownian motionconvergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G57: Random measures
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 657 - 675
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by the National Natural Science Foundation of China (Grant No. 19361060).

References

Bojdecki, T. and Gorostiza, L. G. (1986). Langevin equation for Spr-valued Gaussian processes and fluctuation limits of infinite particle system. Prob. Theory Rel. Fields 73, 227–244.Google Scholar
Dawson, D. A. (1992). Infinitely divisible random measures and superprocesses. In Proceedings of 1990 Workshop on Stochastic Analysis and Related Topics. Silivri, Turkey.Google Scholar
Dawson, D. A. (1993). Measure-valued Markov Processes. In Lecture Notes in Mathematics 1541. Springer, Berlin.Google Scholar
Dawson, D. A. and Ivanoff, D. (1978). Branching diffusions and random measures. Advances in Probability and Related Topics 5, ed. Joffe, A. and Ney, P.. Marcel Dekker, New York, pp. 61–103.Google Scholar
Dynkin, E. B. (1989). Three classes of infinite dimensional diffusion processes. J. Funct. Anal. 86, 75–110.Google Scholar
Dynkin, E. B. (1991). Branching particle systems and superprocesses. Ann. Prob. 19, 1157–1194.Google Scholar
Ethier, S. N. and Kurtz, T.G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
Fitzsimmons, P. J. (1988). Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64, 337–361.Google Scholar
Friedman, A. (1984). Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Gorostiza, L. G. (1988). Limit theorems for supercritical branching random fields with immigration. Adv. Appl. Math. 9, 56–86.Google Scholar
Gorostiza, L. G. and Lopez-Mimbela, J. A. (1990). The multitype measure branching process. Adv. Appl. Prob. 22, 49–67.Google Scholar
Hsu, P. (1986). Brownian exit distribution of a ball. In Seminar on Stochastic Processes, pp. 108–116.Google Scholar
Ivanoff, D. (1981). The branching diffusion with immigration. J. Appl. Prob. 17, 1–15.Google Scholar
Kallenberg, O. (1975). Random Measures. Academic Press, New York.Google Scholar
Li, Z. H. (1991). Integral representations of continuous functions. Chinese Sci. Bull. 36, 979–983.Google Scholar
Li, Z. H. (1992). Measure-valued branching processes with immigration. Stoch. Proc. Appl. 43, 249–264.Google Scholar
Li, Z. H. (1996a). Convolution semigroups associated with measure-valued branching processes. Chinese Sci. Bull. 41, 276–280.Google Scholar
Li, Z. H. (1996b). Immigration structures associated with Dawson–Watanabe superprocesses. Stoch. Proc. Appl. 62, 73–86.Google Scholar
Li, Z. H. and Shiga, T. (1995). Measure-valued branching diffusions: immigrations, excursions and limit theorems. J. Math. Kyoto Univ. 35, 233–274.Google Scholar
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin.Google Scholar
Shiga, T. (1990). A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes. J. Math. Kyoto Univ. 30, 245–279.Google Scholar