Branching processes. I

Vatutin, Vladimir; Zubkov, A. M.

doi:10.1007/bf01086176

Cited by 26 publications

(8 citation statements)

References 296 publications

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“…The continuity theorem (Theorem 9.1) could serve as the basis for a comprehensive theory of long-time scaling limits of critical CSBP, but such a study is beyond the scope of this paper. A large number of results exist in the classical literature that cover supercritical and subcritical cases; see, for example, [2], [18], [25], [27], [40], and [41]. For relatively recent results on critical cases, we refer the reader to [22], [34], and [35].…”

Section: Existence Of Universal Critical Csbpsmentioning

confidence: 99%

Coagulation and universal scaling limits for critical Galton–Watson processes

2018

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The basis of this paper is the elementary observation that the nstep descendant distribution of any Galton-Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain simple necessary and sufficient criteria for the convergence of scaling limits of critical Galton-Watson processes in terms of scaled family-size distributions and a natural notion of convergence of Lévy triples. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain CSBPs satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall in 2006.)Our analysis shows that the nonlinear scaling dynamics of CSBPs becomes linear and purely dilatational when expressed in terms of the Lévy triple associated with the branching mechanism. We prove a continuity theorem for CSBPs in terms of the associated Lévy triples, and use our scaling analysis to prove existence of universal critical Galton-Watson and CSBPs analogous to W. Doeblin's "universal laws". Namely, these universal processes generate all possible critical and subcritical CSBPs as subsequential scaling limits.Our convergence results rely on a natural topology for Lévy triples and a continuity theorem for Bernstein transforms (Laplace exponents) which we develop in a self-contained appendix.

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Section: Existence Of Universal Critical Csbpsmentioning

confidence: 99%

Coagulation and universal scaling limits for critical Galton–Watson processes

2018

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“…Moreover, we will also assume that the Perron root of A, i.e., the maximum positive real eigenvalue of A, is greater than one. Thus we deal with the indecomposable supercritical branching process [39]. Physically, this means that, first, a particle of a given type potentially has descendants, either direct or distant, of any type and, second, the number of particles in the system, on average, progressively increases.…”

Section: Branching Processesmentioning

confidence: 99%

“…Therefore, the mean number of particles in each subsystem increases exponentially. Note that the existence of a positive α is equivalent to the condition that the Perron root of A is greater than one [39,40].…”

Section: Branching Processesmentioning

confidence: 99%

Generalization of the Beck-Cohen superstatistics

Sob’yanin¹

2011

Phys. Rev. E

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Generalized superstatistics, i.e., a "statistics of superstatistics," is proposed. A generalized superstatistical system comprises a set of superstatistical subsystems and represents a generalized hyperensemble. There exists a random control parameter that determines both the density of energy states and the distribution of the intensive parameter for each superstatistical subsystem, thereby forming the third, upper level of dynamics. Generalized superstatistics can be used for nonstationary nonequilibrium systems. The system in which a supercritical multitype age-dependent branching process takes place is an example of a nonstationary generalized superstatistical system. The theory is applied to pair production in a neutron star magnetosphere.

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“…A proper numerical method for the solution of this conflict is not available to the authors's best knowledge. Previous extensive research on the branching of processes, see [31,32], involved stochastic process models, the construction of which is not feasible in the cases considered herein. An attempt for solution is, thus, made following the approach in [18].…”

Section: Introductionmentioning

confidence: 98%

Detection of branching points in noisy processes

Beer

Liebscher²

2009

Comput Mech

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Processes in engineering mechanics often contain branching points at which the system can follow different physical paths. In this paper a method for the detection of these branching points is proposed for processes that are affected by noise. It is assumed that a bundle of process records are available from numerical simulations or from experiments, and branching points are concealed by the noise of the process. The bundle of process records is then evaluated at a series of discrete values of the independent process coordinates. At each discrete point of the process, the associated point set of process values is investigated with the aid of cluster analysis. The detected branching points are verified with a recursive algorithm. The revealed information about the branching points can be used to identify the physical and mechanical background for the branching. This helps to better understand a mechanical system and to design it optimal for a specific purpose. The proposed method is demonstrated by means of both a numerical example and a practical example of a crashworthiness investigation.

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Branching processes. I

Cited by 26 publications

References 296 publications

Coagulation and universal scaling limits for critical Galton–Watson processes

Coagulation and universal scaling limits for critical Galton–Watson processes

Generalization of the Beck-Cohen superstatistics

Detection of branching points in noisy processes

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