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

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:

UDC: 519.218

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
\Bibitem{VatTop11} \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 \mathnet{http://mi.mathnet.ru/mt215} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2961768} \elib{http://elibrary.ru/item.asp?id=17025629} \transl \jour Siberian Adv. Math. \yr 2013 \vol 23 \issue 2 \pages 125--153 \crossref{https://doi.org/10.3103/S1055134413020065} 

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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. Bulinskaya E.V., “Limit Theorems for Local Particle Numbers in Branching Random Walk”, Dokl. Math., 85:3 (2012), 403–405
2. E. Vl. Bulinskaya, “Hitting times with taboo for a random walk”, Siberian Adv. Math., 22:4 (2012), 227–242
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
4. V. A. Vatutin, V. A. Topchii, “Critical Bellman–Harris branching processes with long-living particles”, Proc. Steklov Inst. Math., 282 (2013), 243–272
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
6. Bulinskaya E.V., “Finiteness of Hitting Times Under Taboo”, Stat. Probab. Lett., 85 (2014), 15–19
7. E. Vl. Bulinskaya, “Complete classification of catalytic branching processes”, Theory Probab. Appl., 59:4 (2015), 545–566
8. Bulinskaya E.V., “Local Particles Numbers in Critical Branching Random Walk”, J. Theor. Probab., 27:3 (2014), 878–898
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
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
11. E. Vl. Bulinskaya, “Maksimum kataliticheskogo vetvyaschegosya sluchainogo bluzhdaniya”, UMN, 74:3(447) (2019), 187–188
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