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The Kolmogorov equation in the stochastic fragmentation theory and branching processes with infinite collection of particle types. (English) Zbl 1131.60085

Summary: The stochastic model for the description of the so-called fragmentation process in frameworks of Kolmogorov approach is proposed. This model is represented as the branching process with continuum set \((0,\infty )\) of particle types. Each type \(r\in (0,\infty )\) corresponds to the set of fragments having the size \(r\). It is proved that the branching condition of this process represents the basic equation of the Kolmogorov theory.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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References:

[1] A. F. Filippov, “On the particle size distribution at the subdivision,” Teoriya Veroyatnostei i ee Primenenie, vol. 6, no. 3, pp. 299-318, 1961 (Russian).
[2] A. N. Kolmogorov, “On the logarithmically normal distribution law of particle sizes at the subdivision,” Doklady Akademii Nauk SSSR, vol. 31, no. 2, pp. 99-101, 1941 (Russian).
[3] B. A. Sevast’yanov, Branching Processes, Nauka, Moscow, 1971. · Zbl 0238.60001
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