Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Asymptotic behaviour of measure-valued critical branching processes

Author: Alison M. Etheridge
Journal: Proc. Amer. Math. Soc. 118 (1993), 1251-1261
MSC: Primary 60J80; Secondary 60G57, 60J60
MathSciNet review: 1100650
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Measure-valued branching processes can be characterized in terms of the Laplace transform of their transition densities and this gives rise to a second order nonlinear p.d.e.--the evolution equation of the process. We write the solution to this evolution equation as a series, each of whose coefficients is expressed in terms of the linear semigroup corresponding to the spatial part of the measure-valued process. From this we obtain a simple proof that if the spatial part of the process is a recurrent (resp., transient) Markov process on a standard Borel space and the initial value of the process is an invariant measure of this spatial process, then the process has no (resp., has a unique) nontrivial limiting distribution.

References [Enhancements On Off] (What's this?)

  • [1] E. A. Carlen, S. Kusuoka, and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré 23 (1987), 245-287. MR 898496 (88i:35066)
  • [2] D. A. Dawson, Stochastic evolution equations and related measure processes, J. Multivariate Anal. 5 (1975), 1-52. MR 0388539 (52:9375)
  • [3] -, The critical measure diffusion process, Z. Warsch. Verw. Gebiete 40 (1977), 125-145. MR 0478374 (57:17857)
  • [4] E. B. Dynkin, Superprocesses and their linear additive functionals, Trans. Amer. Math. Soc. 314 (1989), 255-282. MR 930086 (89k:60124)
  • [5] -, Three classes of infinite dimensional diffusions, J. Funct. Anal. 86 (1989), 75-110. MR 1013934 (91b:60061)
  • [6] -, Regular transition functions and regular superprocesses, Trans. Amer. Math. Soc. 316 (1989), 623-634. MR 951884 (90c:60046)
  • [7] S. N. Ethier and T. G. Kurtz, Markov processes: characterization and convergence, Wiley, New York, 1986. MR 838085 (88a:60130)
  • [8] A. M. Etheridge, Asymptotic behaviour of some measure-valued diffusions, Oxford Ph.D. thesis, 1989.
  • [9] -, Measure-valued critical branching diffusion processes with immigration, unpublished manuscript.
  • [10] W. Feller, Diffusion processes in genetics, Proc. 2nd Berkeley Sympos. Math. Statist. Prob., Univ. of California Press, Berkeley, CA, 1951, pp. 227-246. MR 0046022 (13:671c)
  • [11] -, An introduction to probability theory and its applications vol. II, Wiley, New York, 1966. MR 0210154 (35:1048)
  • [12] I. Iscoe, A weighted occupation time for a class of measure-valued branching processes, Probab. Theory Rel. Fields 71 (1986), 85-116. MR 814663 (87c:60070)
  • [13] O. Kallenberg, Stability of critical cluster fields, Math. Nachr. 77 (1975), 7-43. MR 0443078 (56:1451)
  • [14] S. Watanabe, A limit theorem of branching processes and continuous state branching processes, J. Math. Kyoto Univ. 8 (1968), 141-167. MR 0237008 (38:5301)
  • [15] E. Wild, On Boltzmann's equation in the kinetic theory of gases, Math. Proc. Cambridge Philos. Soc. 47 (1951), 602-609. MR 0042999 (13:195e)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60J80, 60G57, 60J60

Retrieve articles in all journals with MSC: 60J80, 60G57, 60J60

Additional Information

Keywords: Measure-valued diffusions, critical branching process, diffusion approximation, asymptotic distribution
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society