Skip to main content
SpringerLink
Log in

Local Particles Numbers in Critical Branching Random Walk

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Critical catalytic branching random walk on an integer lattice ℤd is investigated for all d∈ℕ. The branching may occur at the origin only and the start point is arbitrary. The asymptotic behavior, as time grows to infinity, is determined for the mean local particles numbers. The same problem is solved for the probability of the presence of particles at a fixed lattice point. Moreover, the Yaglom type limit theorem is established for the local number of particles. Our analysis involves construction of an auxiliary Bellman–Harris branching process with six types of particles. The proofs employ the asymptotic properties of the (improper) c.d.f. of hitting times with taboo. The latter notion was recently introduced by the author for a non-branching random walk on ℤd.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albeverio, S., Bogachev, L.V., Yarovaya, E.B.: Asymptotics of branching symmetric random walk on the lattice with a single source. C. R. Acad. Sci. Paris Sér. I Math. 326, 975–980 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bogachev, L.V., Yarovaya, E.B.: The moment analysis of a branching random walk on a lattice with a single source. Dokl. Math. 58, 403–406 (1998)

    Google Scholar 

  3. Bruijn, N.G.: Asymptotic Methods in Analysis. Courier Dover Publications, New York (1981)

    MATH  Google Scholar 

  4. Bulinskaya, E.Vl.: Catalytic branching random walk on three-dimensional lattice. Theory Stoch. Process. 16, 23–32 (2010)

    MATH  MathSciNet  Google Scholar 

  5. Bulinskaya, E.Vl.: Catalytic branching random walk on a two-dimensional lattice. Theory Probab. Appl. 55, 120–126 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bulinskaya, E.Vl.: Limit distributions arising in branching random walks on integer lattices. Lith. Math. J. (2011). doi:10.1007/s10986-011-9128-2

    MathSciNet  Google Scholar 

  7. Bulinskaya, E.Vl.: Limit distributions for the number of particles in branching random walks. Math. Notes 90, 103–116 (2011)

    Article  MathSciNet  Google Scholar 

  8. Bulinskaya, E.Vl.: The hitting times with taboo for a random walk on an integer lattice. Prépublication de LPMA UPMC (univ. Paris-VI) no. 1456 (2011). arXiv:1107.1074v1 [math.PR]

  9. Daletsky Yu, L., Krein, M.G.: Stability of Solution of Differential Equations in Banach Space. Nauka, Moscow (1970) (in Russian)

    Google Scholar 

  10. Gikhman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes II. Springer, Berlin (2004)

    MATH  Google Scholar 

  11. Hu, Y., Vatutin, V., Topchii, V.: Branching random walk in ℤ4 with branching at the origin only (24 Jun 2010). arXiv:1006.4769v1 [math.PR]

  12. Sewastianow, B.A.: Verzweigungsprozesse. Akademie-Verlag, Berlin (1974) (in German)

    Google Scholar 

  13. Topchii, V., Vatutin, V.: Individuals at the origin in the critical catalytic branching random walk. Discrete Math. Theor. Comput. Sci. 6, 325–332 (2003). http://dmtcs.loria.fr/proceedings/html/dmAC7130.abs.html

    MathSciNet  Google Scholar 

  14. Topchii, V.A., Vatutin, V.A.: Catalytic branching random walk in ℤd with branching at the origin only. Sib. Adv. Math. 14, 28–72 (2011)

    MathSciNet  Google Scholar 

  15. Vatutin, V.A.: Limit theorems for critical Markov branching processes with several types of particles and infinite second moments. Math. USSR Sb. 32, 215–225 (1977)

    Article  MATH  Google Scholar 

  16. Vatutin, V.A.: Limit theorem for a critical multitype Bellman–Harris branching process with infinite second moments. Theory Probab. Appl. 23, 776–788 (1979)

    Article  MATH  Google Scholar 

  17. Vatutin, V.A.: Discrete limit distributions of the number of particles in a multitype age-dependent branching processes. Theory Probab. Appl. 24, 509–520 (1980)

    Article  MATH  Google Scholar 

  18. Vatutin, V.A.: On a class of critical Bellman–Harris branching processes with several types of particles. Theory Probab. Appl. 25, 760–771 (1980)

    Article  MathSciNet  Google Scholar 

  19. Vatutin, V.A.: Critical Bellman–Harris branching processes starting with a large number of particles. Math. Notes 40, 803–811 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  20. Vatutin, V.A., Topchii, V.A., Yarovaya, E.B.: Catalytic branching random walk and queueing systems with random number of independent servers. Theory Probab. Math. Stat. 69, 1–15 (2004)

    Article  MathSciNet  Google Scholar 

  21. Yarovaya, E.B.: A limit theorem for critical branching random walk on ℤd with a single source. Russ. Math. Surv. 60, 173–175 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Yarovaya, E.B.: Branching Random Walks in Inhomogeneous Medium. MSU, Moscow (2007) (in Russian)

    Google Scholar 

  23. Yarovaya, E.B.: The monotonicity of the probability of return into the source in models of branching random walks. Mosc. Univ. Math. Bull. 65, 78–80 (2010)

    Article  MathSciNet  Google Scholar 

  24. Yarovaya, E.B.: Criteria of the exponential growth of particles numbers in models of branching random walks. Theory Probab. Appl. 55, 661–682 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work is partially supported by the RFBR grant 10-01-00266.

The author is grateful to Associate Professor E.B. Yarovaya for permanent attention and to Professor V.A. Vatutin for valuable remarks. Special thanks are to Professors I. Kourkova and G. Pagès for invitation to LPMA UPMC (Paris-6) where this work was started.

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Leninskie gory 1, Moscow, 119991, Russia

    Ekaterina Vladimirovna Bulinskaya

Authors
  1. Ekaterina Vladimirovna Bulinskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ekaterina Vladimirovna Bulinskaya.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bulinskaya, E.V. Local Particles Numbers in Critical Branching Random Walk. J Theor Probab 27, 878–898 (2014). https://doi.org/10.1007/s10959-012-0441-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-012-0441-4

Keywords

Mathematics Subject Classification

Search

Navigation