Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-18T21:41:01.876Z Has data issue: false hasContentIssue false
  • English
  • Français

A spatio-temporal point process model for particle growth

Part of: Markov processes Stochastic 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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aiello, W., Bonato, A., Cooper, C., Janssen, J. and Pralat, P. (2008). A spatial web graph model with local influence regions. Internet Math. 5, 173–193.CrossRefGoogle Scholar
Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48.CrossRefGoogle Scholar
Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Banerjee, A., Burlina, P. and Alajiji, F. (1999). Image segmentation and labeling using the Polya urn model. IEEE Trans. Image Proc. 8, 1243–1253.CrossRefGoogle ScholarPubMed
Barabási, A. L. and Albert, R. (1999). Emerging of scaling in random networks. Science 286, 509–512.Google Scholar
Blackwell, D. and Macqueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1, 353–355.CrossRefGoogle Scholar
Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 1–19.CrossRefGoogle Scholar
Durrett, R. (2008). Probability Models of DNA Sequence Evolution, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketteter Vorgänge. Z. Angew. Math. Mech. 3, 279–289.CrossRefGoogle Scholar
Evans, S. N. and Pitman, J. (1998). Construction of Markovian coalescents. Ann. Inst. Henri Poincaré 34, 339–383.CrossRefGoogle Scholar
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87–112.CrossRefGoogle ScholarPubMed
Haccou, P., Jagers, P. and Vatutin, V. A. (2005). Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.CrossRefGoogle Scholar
Hayhoe, M., Alajiji, F. and Gharesifard, B. (2018). A Polya contagion model for networks. IEEE Trans. Control Network Systems 5, 1998–2010.CrossRefGoogle Scholar
Herbots, H. M. (1997). The structured coalescent. In Progress in Population Genetics and Human Evolution, eds Donnelly, P. and Tavaré, S., Springer, New York, pp. 231–255.CrossRefGoogle Scholar
Hoppe, F. M. (1984). Pólya-like urns and the Ewens sampling formula. J. Math. Biol. 20, 91–94.CrossRefGoogle Scholar
Hoppe, F. M. (1987). The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25, 123–159.CrossRefGoogle ScholarPubMed
Jacob, E. and Mörters, P. (2015). Spatial preferential attachment networks: power laws and clustering coefficients. Ann. Appl. Prob. 25, 632–662.CrossRefGoogle Scholar
Jacobsen, M. (2005). Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Birkhäuser, Boston.Google Scholar
Joyce, P. and Tavaré, S. (1987). Cycles, permutations and the structure of the Yule process with immigration. Stoch. Process. Appl. 25, 309–314.CrossRefGoogle Scholar
Kendall, D. G. (1949). Stochastic processes and population growth. J. Roy. Statist. Soc. B 11, 230–264.Google Scholar
Kimura, M. and Crow, J. F. (1964). The number of alleles that can be maintained in a finite population. Genetics 49, 725–738.Google Scholar
Kingman, J. F. C. (1975). Random discrete distributions. J. Roy. Statist. Soc. Ser. B 37, 1–22.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
Nehring, B. and Zessin, H. (2011). Der Papangelou prozess: A concept for Gibb, Fermi and Bose processes. J. Contemp. Math. Anal. 44, 36–44.Google Scholar
Notohara, M. (1990). The coalescent and the genealogical process in geographically structured populations. J. Math. Biol. 29, 59–75.CrossRefGoogle Scholar
Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855–900.Google Scholar
Rafler, M. (2012). Hoppe trees, random recursive sets and their barycenter. Preprint. Available at https://arxiv.org/abs/1207.1636v1.Google Scholar
Tavaré, S. (1987). The birth process with immigration, and the genealogical structure of of large populations. J. Math. Biol. 25, 161–168.CrossRefGoogle ScholarPubMed
Watterson, G. A. (1984). Lines of descent and the coalescent. Theoret. Pop. Biol. 26, 77–92.CrossRefGoogle Scholar