Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T01:45:53.266Z Has data issue: false hasContentIssue false
  • English
  • Français

An infinite particle system with additive interactions

Published online by Cambridge University Press:  01 July 2016

Richard Durrett
Richard Durrett*
Affiliation:
University of California, Los Angeles
*
∗Postal address: Department of Mathematics, University of California, Los Angeles CA 90024, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The models under consideration are a class of infinite particle systems which can be written as a superposition of branching random walks. This paper gives some results about the limiting behavior of the number of particles in a compact set as t → ∞ and also gives both sufficient and necessary conditions for the existence of a non-trivial translation-invariant stationary distribution.

Keywords

INFINITE PARTICLE SYSTEMRANDOM WALKBRANCHING PROCESSPOINT PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 2 , June 1979 , pp. 355 - 383
Copyright
Copyright © Applied Probability Trust 1979 

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.)

References

Asmussen, S. and Kaplan, N. (1976) Branching random walks I. Stock. Proc. Appl. 4, 1–13.CrossRefGoogle Scholar
Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
Buhler, W. J. (1970) The distribution of generations and other aspects of the family structure of a branching process. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 463–480.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of point processes. In Lewis (1972), 463–480.Google Scholar
Debes, H., Kerstan, J., Liemant, A. and Matthes, K. (1970) Verallgemeinerungen eines Satzes von Dobrushin, I. Math. Nachr. 47, 183–224.CrossRefGoogle Scholar
Dobrushin, R. L. (1956) On Poisson laws for the distribution of particles in space. Ukrain. Math. Z. 8, 127–134.Google Scholar
Doob, J. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Dunford, N. (1951) An individual ergodic theorem for noncommutative transformations. Acta Sci. Math. (Szeged) 14, 1–14.Google Scholar
Fleischman, J. (1978) Limiting distributions for branching random fields. Trans. Amer. Math. Soc. 239, 353–390.Google Scholar
Harris, T. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Harris, T. (1976) On a class of set-valued Markov processes. Ann. Prob. 4, 175–194.CrossRefGoogle Scholar
Harris, T. (1978) Additive set-valued Markov processes and percolation methods. Ann. Prob. 6, 355–378.Google Scholar
Hille, E. (1962) Analytic Function Theory, Vol. II. Ginn, New York.Google Scholar
Holley, R. and Strook, D. (1978) Nearest neighbour birth and death processes on the real line. Acta Math. 140, 103–154.Google Scholar
Jagers, P. (1974) Aspects of random measures and point processes. Advances in Probability 3, ed. Ney, P. and Port, S., Dekker, New York, 179–239.Google Scholar
Joffe, A. and Moncayo, A. (1973) Random variables trees and branching random walks. Adv. Maths 10, 401–416.Google Scholar
Kallenberg, O. (1977) Stability of critical cluster fields. Math. Nachr. 77, 7–43.Google Scholar
Kaplan, N. and Asmussen, S. (1976) Branching random walks II. Stoch. Proc. Appl. 4, 15–31.Google Scholar
Kharmalov, B. P. (1968) On properties of branching processes with an arbitrary set of particle types. Theory Prob. Appl. 13, 84–98.Google Scholar
Lewis, P. A. W., (ed.) (1972) Stochastic Point Processes: Statistical Analysis, Theory, and Applications. Wiley, New York.Google Scholar
Matthes, K. (1972) Infinitely divisible point processes. In Lewis (1972), 384–404.Google Scholar
Ney, P. (1965) The convergence of a random distribution function associated with a branching process. J. Math. Anal. Appl. 12, 316–327.CrossRefGoogle Scholar
Samuels, M. L. (1971) Distribution of the branching process population among generations. J. Appl. Prob. 8, 655–667.Google Scholar
Sawyer, S. (1976) Branching diffusion processes in population genetics. Adv. Appl. Prob. 8, 659–689.CrossRefGoogle Scholar
Smythe, R. (1976) Multiparameter subadditive processes. Ann. Prob. 4, 772–782.CrossRefGoogle Scholar
Stone, C. (1965) On local and ratio limit theorems. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2, 217–224.Google Scholar
Stone, C. (1968) On a theorem of Dobrushin. Ann. Math. Statist. 39, 1391–1401.Google Scholar
Sudbury, A. (1977) Clumping effects in models of isolation by distance. J. Appl. Prob. 14, 391–395.Google Scholar