The Annals of Applied Probability

Stochastic coagulation-fragmentation processes with a finite number of particles and applications

Nathanael Hoze and David Holcman

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Coagulation-fragmentation processes describe the stochastic association and dissociation of particles in clusters. Cluster dynamics with cluster-cluster interactions for a finite number of particles has recently attracted attention especially in stochastic analysis and statistical physics of cellular biology, as novel experimental data are now available, but their interpretation remains challenging. We derive here probability distribution functions for clusters that can either aggregate upon binding to form clusters of arbitrary sizes or a single cluster can dissociate into two sub-clusters. Using combinatorics properties and Markov chain representation, we compute steady-state distributions and moments for the number of particles per cluster in the case where the coagulation and fragmentation rates follow a detailed balance condition. We obtain explicit and asymptotic formulas for the cluster size and the number of clusters in terms of hypergeometric functions. To further characterize clustering, we introduce and discuss two mean times: one is the mean time two particles spend together before they separate and the other is the mean time they spend separated before they meet again for the first time. Finally, we discuss applications of the present stochastic coagulation-fragmentation framework in cell biology.

Article information

Ann. Appl. Probab., Volume 28, Number 3 (2018), 1449-1490.

Received: November 2016
Revised: May 2017
First available in Project Euclid: 1 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
Secondary: 05A17: Partitions of integers [See also 11P81, 11P82, 11P83]

Markov chain coagulation-fragmentation processes stochastic processes


Hoze, Nathanael; Holcman, David. Stochastic coagulation-fragmentation processes with a finite number of particles and applications. Ann. Appl. Probab. 28 (2018), no. 3, 1449--1490. doi:10.1214/17-AAP1334.

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