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Mat. Tr., 2011, Volume 14, Number 2, Pages 28–72 (Mi mt215)  

This article is cited in 11 scientific papers (total in 11 papers)

Catalytic branching random walks in $\mathbb Z^d$ with branching at the origin

V. A. Vatutina, V. A. Topchiĭb

a Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
b Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Science, Omsk, Russia

Abstract: A time-continuous branching random walk on the lattice $\mathbb Z^d$, $d\ge1$, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment $t=0$ by a single particle located at the origin, and the average number of offspring produced at the origin is such that the corresponding branching random walk is critical. The asymptotic behavior of the survival probability of such a process at moment $t\to\infty$ and the presence of at least one particle at the origin is studied. In addition, we obtain the asymptotic expansions for the expectation of the number of particles at the origin and prove Yaglom-type conditional limit theorems for the number of particles located at the origin and beyond at moment $t$.

Key words: catalytic branching random walk, a homogeneous and symmetric time-continuous multidimensional Markov random walk, Bellman–Harris branching process with two types of particles, renewal theory, limit theorem.

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English version:
Siberian Advances in Mathematics, 2013, 23:2, 125–153

Bibliographic databases:

Document Type: Article
UDC: 519.218
Received: 27.04.2010

Citation: V. A. Vatutin, V. A. Topchiǐ, “Catalytic branching random walks in $\mathbb Z^d$ with branching at the origin”, Mat. Tr., 14:2 (2011), 28–72; Siberian Adv. Math., 23:2 (2013), 125–153

Citation in format AMSBIB
\by V.~A.~Vatutin, V.~A.~Topchi{\v\i}
\paper Catalytic branching random walks in~$\mathbb Z^d$ with branching at the origin
\jour Mat. Tr.
\yr 2011
\vol 14
\issue 2
\pages 28--72
\jour Siberian Adv. Math.
\yr 2013
\vol 23
\issue 2
\pages 125--153

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    This publication is cited in the following articles:
    1. Bulinskaya E.V., “Limit Theorems for Local Particle Numbers in Branching Random Walk”, Dokl. Math., 85:3 (2012), 403–405  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    2. E. Vl. Bulinskaya, “Hitting times with taboo for a random walk”, Siberian Adv. Math., 22:4 (2012), 227–242  mathnet  crossref  mathscinet  elib
    3. V. A. Topchii, “The asymptotic behaviour of derivatives of the renewal function for distributions with infinite first moment and regularly varying tails of index $\beta\in(1/2,1]$”, Discrete Math. Appl., 22:3 (2012), 315–344  mathnet  crossref  crossref  mathscinet  elib
    4. V. A. Vatutin, V. A. Topchii, “Critical Bellman–Harris branching processes with long-living particles”, Proc. Steklov Inst. Math., 282 (2013), 243–272  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. E. Vl. Bulinskaya, “Subcritical catalytic branching random walk with finite or infinite variance of offspring number”, Proc. Steklov Inst. Math., 282 (2013), 62–72  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. Bulinskaya E.V., “Finiteness of Hitting Times Under Taboo”, Stat. Probab. Lett., 85 (2014), 15–19  crossref  mathscinet  zmath  isi  elib  scopus
    7. E. Vl. Bulinskaya, “Complete classification of catalytic branching processes”, Theory Probab. Appl., 59:4 (2015), 545–566  mathnet  crossref  crossref  mathscinet  isi  elib
    8. Bulinskaya E.V., “Local Particles Numbers in Critical Branching Random Walk”, J. Theor. Probab., 27:3 (2014), 878–898  crossref  zmath  isi  elib  scopus
    9. Bulinskaya E.V., “Strong and Weak Convergence of the Population Size in a Supercritical Catalytic Branching Process”, Dokl. Math., 92:3 (2015), 714–718  crossref  zmath  isi  elib  scopus
    10. V. A. Topchiǐ, “On renewal matrices connected with branching processes with tails of distributions of different orders”, Siberian Adv. Math., 28:2 (2018), 115–153  mathnet  crossref  crossref  elib
    11. E. Vl. Bulinskaya, “Maksimum kataliticheskogo vetvyaschegosya sluchainogo bluzhdaniya”, UMN, 74:3(447) (2019), 187–188  mathnet  crossref
  • Математические труды Siberian Advances in Mathematics
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