Branching processes with lattice spatial dynamics and a finite set of particle generation centers

Molchanov, Stanislav; Yarovaya, Elena

doi:10.1134/s1064562412040278

Cited by 27 publications

(28 citation statements)

References 6 publications

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“…Remark that in [27] there is also used the concept of the propagation front of CBRW with binary fission and symmetric random walk on Z d , namely, Γ t = y = y(t) ∈ Z d : E 0 µ(t; y) < C where C is some positive constant. Moreover, in the framework of our terminology there is shown that Γ t = t (P ∪ O).…”

Section: Notation Main Results and Discussionmentioning

confidence: 99%

Spread of a catalytic branching random walk on a multidimensional lattice

Bulinskaya

2018

Stochastic Processes and their Applications

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For a supercritical catalytic branching random walk on Z d , d ∈ N, with an arbitrary finite catalysts set we study the spread of particles population as time grows to infinity. Namely, we divide by t the position coordinates of each particle existing at time t and then let t tend to infinity. It is shown that in the limit there are a.s. no particles outside the closed convex surface in R d which we call the propagation front and, under condition of infinite number of visits of the catalysts set, a.s. there exist particles on the propagation front. We also demonstrate that the propagation front is asymptotically densely populated and derive its alternative representation. Recent strong limit theorems for total and local particles numbers established by the author play an essential role. The results obtained develop ones by Ph.Carmona and Y.Hu (2014) devoted to the spread of catalytic branching random walk on Z.

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Section: Notation Main Results and Discussionmentioning

confidence: 99%

Spread of a catalytic branching random walk on a multidimensional lattice

Bulinskaya

2018

Stochastic Processes and their Applications

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“…where G * 1 (ν) = β 1 / (ν + β 1 ). Hence, taking into account definition (13) and operating similarly to deriving the asymptotic behavior of the function J 3 (t), as t → ∞, we get relation (25). Lemma 3 is proved completely.…”

Section: Lemma 2 Let Conditionsmentioning

confidence: 80%

“…The main subjects of study were the total and local particles numbers at time t and their behavior, as t → ∞. In 2012 in the paper [25] for the first time there was set and solved the problem of the spread of particles population in supercritical CBRW, when the symmetric random walk has "light" tails and the particles reproduction is binary. There was also studied the rate of movement of the population propagation front, determined in terms of the moments boundedness of the local particles numbers, as t → ∞.…”

Section: Introductionmentioning

confidence: 99%

Fluctuations of the Propagation Front of a Catalytic Branching Walk

Bulinskaya¹

2020

Theory Probab. Appl.

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We consider a supercritical catalytic branching random walk (CBRW) on a multidimensional lattice Z d , d ∈ N. The main subject of study is the behavior of particles cloud in space and time. For CBRW on an integer line, Carmona and Hu (2014) examined the asymptotical behavior of the maximal coordinate M n of the particles at time n. They proved that M n /n → µ almost surely (on a set of local non-degeneracy of CBRW), as n → ∞, where µ > 0 is a certain constant. Under additional assumption of a single catalyst in CBRW they also investigated the fluctuations of M n with respect to µn, as n → ∞. Bulinskaya (2018) extended the strong limit theorem by Carmona and Hu having estimated the rate of the population propagation for the front of a multidimensional CBRW. Now our aim is to analyze fluctuations of the propagation front in CBRW on Z d . We not only solve the problem in a multidimensional setting but also, treating the case of an arbitrary finite number of catalysts for d = 1, generalize the result by Carmona and Hu with the help of other probabilistic-analytic methods.

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“…random variables with the same distribution as ζ. According to strong law of large numbers and bounded convergence theorem, inequality (21) implies that 0 ≤ K(λ) ≤ K(0+).…”

Section: Lemma 7 If Conditionsmentioning

confidence: 99%

“…There were applied the new classification of CBRW introduced in [8] and limit theorems for local and total particles numbers established in [9]. It is worth mentioning that related results in terms of boundedness of some random variables moments, rather than strong or weak convergence of maximum of CBRW, were obtained in a series of papers, see, e.g., [21]. Furthermore, various characteristics of particles propagation were analyzed in the framework of spatially continuous counterpart of CBRW called catalytic branching Brownian motion, see, e.g., [3] and [29].…”

Section: Introductionmentioning

confidence: 99%

Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

Bulinskaya

2020

J Theor Probab

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For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a nontrivial law. An explicit formula is provided for this normalization and non-linear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with "heavy" tails, in contrast to the known behavior in case of "light" tails of the random walk jumps. The new tools such as "many-to-few lemma" and spinal decomposition appear non-efficient here. The approach developed in the paper combines the techniques of renewal theory, Laplace transform, non-linear integral equations and large deviations theory for random sums of random variables.

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Branching processes with lattice spatial dynamics and a finite set of particle generation centers

Cited by 27 publications

References 6 publications

Spread of a catalytic branching random walk on a multidimensional lattice

Spread of a catalytic branching random walk on a multidimensional lattice

Fluctuations of the Propagation Front of a Catalytic Branching Walk

Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

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