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On the Number of Positive Eigenvalues of the Evolutionary Operator of Branching Random Walk

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Branching Processes and Their Applications

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 219))

Abstract

We consider a continuous-time branching random walk on a multidimensional lattice with finite variance of jumps and a finite set of the particle generation centers, i.e. branching sources. The main object of interest is the evolutionary operator for the mean number of particles both at an arbitrary point and on the entire lattice. It is shown that the amount of its positive eigenvalues, counting their multiplicity, does not exceed the amount of branching sources on the lattice, while the maximal of these eigenvalues is always simple. We present also an example demonstrating that the symmetry of the spatial configuration of sources can lead to appearance of multiple lower eigenvalues in the spectrum of the evolutionary operator.

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References

  1. Albeverio, S., Bogachev, L.: Branching random walk in a catalytic medium. Basic Equ. (2000). doi:10.1023/A:1009818620550

    MathSciNet  MATH  Google Scholar 

  2. Albeverio, S., Bogachev, L., Yarovaya, E.: Asymptotics of branching symmetric random walk on the lattice with a single source. C. R. Acad. Sci. Paris Sér. I Math. (1998). doi:10.1016/S0764-4442(98)80125-0

    MATH  Google Scholar 

  3. Bogachev, L., Yarovaya, E.: A limit theorem for a supercritical branching random walk on \(\mathbb{Z}^{d}\) with a single source. Russ. Math. Surv. 53 (5), 1086–1088 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Daleckii, J., Krein, M.: Stability of Solutions of Differential Equations in Banach Space. AMS, Providence (1974)

    Google Scholar 

  5. Gantmacher, F.: Theory of Matrices. AMS Chelsea Publishing, New York (1959)

    MATH  Google Scholar 

  6. Gikhman, I., Skorokhod, A.: Theory of Stochastic Processes. Springer, New York (2004–2007)

    MATH  Google Scholar 

  7. Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, New York (1985)

    Book  MATH  Google Scholar 

  8. Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966)

    Book  MATH  Google Scholar 

  9. Molchanov, S., Yarovaya, E.: The population structure inside the propagation front of a branching random walk with a finite number of particle generation centers. Dokl. Math. 86 (3), 787–790 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Vatutin, V., Topchiĭ, V.: Limit theorem for critical catalytic branching random walks. Theory Probab. Appl. 49 (3), 498–518 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Vatutin, V., Topchiĭ, V., Yarovaya, E.: Catalytic branching random walks and queueing systems with a random number of independent servers. Theory Probab. Math. Stat. 69, 1–15 (2004)

    Article  Google Scholar 

  12. Yarovaya, E.: Branching Random Walks in an Inhomogeneous Medium. Center of Applied Investigations of the Faculty of Mechanics and Mathematics of the Moscow State University, Moscow (2007) [in Russian]

    Google Scholar 

  13. Yarovaya, E.: About the investigation of branching random walks on multidimensional lattices. Sovrem. Probl. Mat. Mech. Mosc. 4, 119–136 (2009) [in Russian]

    Google Scholar 

  14. Yarovaya, E.: Criteria of exponential growth for the numbers of particles in models of branching random walks. Theory Probab. Appl. 55 (4), 661–682 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yarovaya, E.: Spectral properties of evolutionary operators in branching random walk models. Math. Notes 92 (1), 124–140 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Yarovaya, E.: Branching random walks with several sources. Math. Popul. Stud. 20 (1), 14–26 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This study was performed at Lomonosov Moscow state University and at Steklov Mathematical Institute, Russian Academy of Sciences. The work was supported by the Russian Science Foundation, project no. 14-21-00162.

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Authors and Affiliations

  1. Department of Probability Theory, Lomonosov Moscow State University, Leninskie Gory, Main Building, Moscow, Russia

    Ekaterina Antonenko & Elena Yarovaya

Authors
  1. Ekaterina Antonenko
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  2. Elena Yarovaya
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Corresponding author

Correspondence to Elena Yarovaya .

Editor information

Editors and Affiliations

  1. Departamento de Matemáticas Facultad de Ciencias, Universidad de Extremadura, Badajoz, Spain

    Inés M. del Puerto

  2. Departamento de Matemáticas Facultad de Ciencias, Universidad de Extremadura, Badajoz, Spain

    Miguel González

  3. Departamento de Matemáticas Facultad de Ciencias, Universidad de Extremadura, Badajoz, Spain

    Cristina Gutiérrez

  4. Departamento de Matemáticas Centro Universitario de Plasencia, Universidad de Extremadura, Plasencia, Spain

    Rodrigo Martínez

  5. Departamento de Matemáticas Facultad de Ciencias, Universidad de Extremadura, Badajoz, Spain

    Carmen Minuesa

  6. Departamento de Matemáticas Facultad de Ciencias, Universidad de Extremadura, Badajoz, Spain

    Manuel Molina

  7. Departamento de Matemáticas Facultad de Ciencias, Universidad de Extremadura, Badajoz, Spain

    Manuel Mota

  8. Departamento de Matemáticas Facultad de Veterinaria, Universidad de Extremadura, Cáceres, Spain

    Alfonso Ramos

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Antonenko, E., Yarovaya, E. (2016). On the Number of Positive Eigenvalues of the Evolutionary Operator of Branching Random Walk. In: del Puerto, I., et al. Branching Processes and Their Applications. Lecture Notes in Statistics(), vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-31641-3_3

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