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Mat. Zametki, 2012, Volume 92, Issue 1, Pages 123–140 (Mi mz9485)  

This article is cited in 21 scientific papers (total in 21 papers)

Spectral Properties of Evolutionary Operators in Branching Random Walk Models

E. B. Yarovaya

M. V. Lomonosov Moscow State University

Abstract: We introduce a model of continuous-time branching random walk on a finite-dimensional integer lattice with finitely many branching sources of three types and study the spectral properties of the operator describing the evolution of the average numbers of particles both at an arbitrary source and on the entire lattice. For the leading positive eigenvalue of the operator, we obtain existence conditions determining exponential growth in the number of particles in this model.

Keywords: branching random walk, equations in Banach spaces, pseudodifference operator, symmetrizable operator, positive eigenvalue

DOI: https://doi.org/10.4213/mzm9485

Full text: PDF file (637 kB)
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English version:
Mathematical Notes, 2012, 92:1, 115–131

Bibliographic databases:

UDC: 517.984.5+519.21
Received: 05.04.2011
Revised: 21.04.2011

Citation: E. B. Yarovaya, “Spectral Properties of Evolutionary Operators in Branching Random Walk Models”, Mat. Zametki, 92:1 (2012), 123–140; Math. Notes, 92:1 (2012), 115–131

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. A. Molchanov, E. B. Yarovaya, “Limit theorems for the Green function of the lattice Laplacian under large deviations of the random walk”, Izv. Math., 76:6 (2012), 1190–1217  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. S. A. Molchanov, E. B. Yarovaya, “Population structure inside the propagation front of a branching random walk with finitely many centers of particle generation”, Dokl. Math., 86:3 (2012), 787–790  crossref  mathscinet  zmath  isi  elib
    3. S. A. Molchanov, E. B. Yarovaya, “Branching processes with lattice spatial dynamics and a finite set of particle generation centers”, Dokl. Math., 86:2 (2012), 638–641  crossref  mathscinet  zmath  isi  elib  elib
    4. S. A. Molchanov, E. B. Yarovaya, “Large deviations for a symmetric branching random walk on a multidimensional lattice”, Proc. Steklov Inst. Math., 282 (2013), 186–201  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. E. B. Yarovaya, “Operators satisfying the Schur condition and their applications to the branching random walks”, Comm. Statist. Theory Methods, 43:7 (2014), 1523–1532  crossref  mathscinet  zmath  isi  elib
    6. E. V. Bulinskaya, “Strong and weak convergence of the population size in a supercritical catalytic branching process”, Dokl. Math., 92:3 (2015), 714–718  crossref  mathscinet  zmath  isi
    7. E. B. Yarovaya, “The structure of the positive discrete spectrum of the evolution operator arising in branching random walks”, Dokl. Math., 92:1 (2015), 507–510  mathnet  crossref  mathscinet  zmath  isi  elib
    8. E. A. Antonenko, “A weakly supercritical mode in a branching random walk”, Mosc. Univ. Math. Bull., 71:2 (2016), 68–70  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    9. E. Antonenko, E. Yarovaya, “On the number of positive eigenvalues of the evolutionary operator of branching random walk”, Branching Processes and Their Applications, Lecture Notes in Statistics, 219, eds. I. Del Puerto, M. Gonzalez, C. Gutierrez, R. Martinez, C. Minuesa, M. Molina, M. Mota, A. Ramos, Springer, 2016, 41–55  crossref  mathscinet  zmath  isi
    10. E. B. Yarovaya, “Spectral asymptotics of supercritical branching random process”, Theory Probab. Appl., 62:3 (2018), 413–431  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. E. B. Yarovaya, “Positive discrete spectrum of the evolutionary operator of supercritical branching walks with heavy tails”, Methodol. Comput. Appl. Probab., 19:4 (2017), 1151–1167  crossref  mathscinet  zmath  isi
    12. S. Molchanov, J. Whitmeyer, “Stationary cistributions in Kolmogorov-Petrovski-Piskunov-type models with an infinite number of particles”, Math. Popul. Stud., 24:3 (2017), 147–160  crossref  mathscinet  isi
    13. M. V. Platonova, K. S. Ryadovkin, “Asimptoticheskoe povedenie srednego chisla chastits vetvyaschegosya sluchainogo bluzhdaniya na reshetke $\mathbf Z^d$ s periodicheskimi istochnikami vetvleniya”, Veroyatnost i statistika. 26, Zap. nauchn. sem. POMI, 466, POMI, SPb., 2017, 234–256  mathnet
    14. M. V. Platonova, K. S. Ryadovkin, “On the mean number of particles of a branching random walk on $\mathbb{Z}^d$ with periodic sources of branching”, Dokl. Math., 97:2 (2018), 140–143  crossref  mathscinet  zmath  isi
    15. E. B. Yarovaya, “Branching random walk with receding sources”, Russian Math. Surveys, 73:3 (2018), 549–551  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    16. K. S. Ryadovkin, “Asimptoticheskoe povedenie vetvyaschikhsya sluchainykh bluzhdanii na nekotorykh dvumernykh reshetkakh”, Veroyatnost i statistika. 27, Zap. nauchn. sem. POMI, 474, POMI, SPb., 2018, 213–221  mathnet
    17. Ermishkina E., Yarovaya E., “Simulation of Stochastic Processes With Generation and Transport of Particles”, Statistics and Simulation, Iws 8 2015, Springer Proceedings in Mathematics & Statistics, 231, eds. Pilz J., Rasch D., Melas V., Moder K., Springer, 2018, 129–143  crossref  mathscinet  zmath  isi  scopus
    18. E. M. Ermishkina, E. B. Yarovaya, “Modelirovanie vetvyaschikhsya sluchainykh bluzhdanii po mnogomernoi reshëtke”, Fundament. i prikl. matem., 22:3 (2018), 37–56  mathnet
    19. “Abstracts of talks given at the 3rd International Conference on Stochastic Methods”, Theory Probab. Appl., 64:1 (2019), 124–169  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    20. M. V. Platonova, K. S. Ryadovkin, “Branching random walks on $\mathbf{Z}^d$ with periodic branching sources”, Theory Probab. Appl., 64:2 (2019), 229–248  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    21. I. Khristolyubov, E. B. Yarovaya, “A limit theorem for supercritical random branching walks with branching sources of varying intensity”, Theory Probab. Appl., 64:3 (2019), 365–384  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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