Maximal displacement in a branching random walk through interfaces

Mallein, Bastien

doi:10.1214/ejp.v20-2828

Cited by 37 publications

(35 citation statements)

References 31 publications

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“…It is interesting to note the differences in the behavior of the maximum M n in a standard BRW on Z (see, e.g., [24]) and in CBRW considered in the present work. In both models the main term of the asymptotic behavior of M n is a linear function of time n. Besides the linear term in the representation of M n , BRW has a negative logarithmic term and stochastically bounded fluctuations, whereas in the case of CBRW there are only stochastically bounded fluctuations.…”

Section: Introductionmentioning

confidence: 85%

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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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Section: Introductionmentioning

confidence: 85%

“…To estimate the integral J 2 (t) apply inequality (14), when s = r + ε 0 for some ε 0 > 0, and also Theorem 25 in book [30], p. 30. As a result we have (24) as t → ∞, when v(t) = o(t). Combination of the established formulae (21), (23) and (24) implies the desired relation (20).…”

Section: Lemma 2 Let Conditionsmentioning

confidence: 95%

Fluctuations of the Propagation Front of a Catalytic Branching Walk

Bulinskaya¹

2020

Theory Probab. Appl.

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“…where ζ is a random variable with G as its c.d.f. Combination of relations (17)- (20), for t > 2T , leads to the inequality…”

Section: Lemma 7 If Conditionsmentioning

confidence: 99%

“…The rate of the population propagation in this case depends essentially on the conditions imposed on "walking" and "branching". Recent study of maximum of particles positions in the standard space-homogeneous BRW on real line was carried out in [16], [18] and [20] under condition of "light" tails of the distribution of the random walk jump. The case of "heavy" tails required other handling provided, e.g., in [2], [13] and [17].…”

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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“…Strictly speaking,[40] also requires a finite branching ( Z t (R) < ∞ a.s.), but this condition turns out to be unnecessary (see e.g [34][1] that we invoke to prove (b), and the conclusion of (b) obviously implies (a)).…”

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confidence: 99%

Asymptotics of self-similar growth-fragmentation processes

Dadoun¹

2017

Electron. J. Probab.

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Markovian growth-fragmentation processes introduced in [8,9] extend the pure-fragmentation model by allowing the fragments to grow larger or smaller between dislocation events. What becomes of the known asymptotic behaviors of self-similar pure fragmentations [6,11,12,14] when growth is added to the fragments is a natural question that we investigate in this paper. Our results involve the terminal value of some additive martingales whose uniform integrability is an essential requirement. Dwelling first on the homogeneous case [8], we exploit the connection with branching random walks and in particular the martingale convergence of Biggins [18,19] to derive precise asymptotic estimates. The self-similar case [9] is treated in a second part; under the so called Malthusian hypotheses and with the help of several martingale-flavored features recently developed in [10], we obtain limit theorems for empirical measures of the fragments.

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Maximal displacement in a branching random walk through interfaces

Cited by 37 publications

References 31 publications

Fluctuations of the Propagation Front of a Catalytic Branching Walk

Fluctuations of the Propagation Front of a Catalytic Branching Walk

Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

Asymptotics of self-similar growth-fragmentation processes

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