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A spatio-temporal point process model for particle growth

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  12 July 2019

Ola Hössjer
Ola Hössjer*
Affiliation:
Stockholm University
*
*Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden. Email address: ola@math.su.se
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Abstract

A spatio-temporal model of particle or star growth is defined, whereby new unit masses arrive sequentially in discrete time. These unit masses are referred to as candidate stars, which tend to arrive in mass-dense regions and then either form a new star or are absorbed by some neighbouring star of high mass. We analyse the system as time increases, and derive the asymptotic growth rate of the number of stars as well as the size of a randomly chosen star. We also prove that the size-biased mass distribution converges to a Poisson–Dirichlet distribution. This is achieved by embedding our model into a continuous-time Markov process, so that new stars arrive according to a marked Poisson process, with locations as marks, whereas existing stars grow as independent Yule processes. Our approach can be interpreted as a Hoppe-type urn scheme with a spatial structure. We discuss its relevance for and connection to models of population genetics, particle aggregation, image segmentation, epidemic spread, and random graphs with preferential attachment.

Keywords

Hoppe-type urn schememarked point processmass distributionPoisson–Dirichlet distributionspatio-temporal processYule process

MSC classification

Primary: 60G55: Point processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 23 - 38
Copyright
© Applied Probability Trust 2019 

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