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Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes

Part of: Markov processes Special processes

Published online by Cambridge University Press:  20 March 2018

Viktor Bezborodov ,
Luca Di Persio ,
Tyll Krueger ,
Mykola Lebid  and
Tomasz Ożański
Viktor Bezborodov*
Affiliation:
University of Verona
Luca Di Persio*
Affiliation:
University of Verona
Tyll Krueger*
Affiliation:
University of Wrocław
Mykola Lebid*
Affiliation:
ETH Zürich
Tomasz Ożański*
Affiliation:
University of Wrocław
*
* Postal address: Department of Computer Science, The University of Verona, Strada le Grazie 15, Verona, 37134, Italy.
* Postal address: Department of Computer Science, The University of Verona, Strada le Grazie 15, Verona, 37134, Italy.
*** Postal address: Department of Computer Science and Engineering, Wrocław University of Technology, Janiszewskiego 15, Wrocław, 50-372, Poland.
**** Postal address: Department of Biosystems Science and Engineering, ETH Zürich, D-BSSE, Mattenstrasse 26, Basel, 4058, Switzerland.
*** Postal address: Department of Computer Science and Engineering, Wrocław University of Technology, Janiszewskiego 15, Wrocław, 50-372, Poland.
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Abstract

We formulate and prove a shape theorem for a continuous-time continuous-space stochastic growth model under certain general conditions. Similar to the classical lattice growth models, the proof makes use of the subadditive ergodic theorem. A precise expression for the speed of propagation is given in the case of a truncated free-branching birth rate.

Keywords

Shape theoremspatial birth processgrowth model

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 74 - 101
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1] Auffinger, A., Damron, M. and Hanson, J. (2017). 50 Years of First-Passage Percolation (University Lecture Series 68). American Mathematical Society. Google Scholar
[2] Bezborodov, V. (2015). Spatial birth-and-death Markov dynamics of finite particle systems. Preprint. Available at https://arxiv.org/abs/1507.05804. Google Scholar
[3] Biggins, J. D. (1995). The growth and spread of the general branching random walk. Ann. Appl. Prob. 5, 1008–1024. Google Scholar
[4] Burke, C. J. and Rosenblatt, M. (1958). A Markovian function of a Markov chain. Ann. Math. Statist. 29, 1112–1122. Google Scholar
[5] Deijfen, M. (2003). Asymptotic shape in a continuum growth model. Adv. Appl. Prob. 35, 303–318. Google Scholar
[6] Durrett, R. (1983). Maxima of branching random walks. Z. Wahrscheinlichkeitsth. 62, 165–170. Google Scholar
[7] Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole, Pacific Grove, CA. Google Scholar
[8] Eden, M. (1961). A two-dimensional growth process. In Proc. 4th Berkeley Symp. Math. Statist. Prob., Vol. IV, University of California Press, Berkeley, CA, pp. 223–239. Google Scholar
[9] Eibeck, A. and Wagner, W. (2003). Stochastic interacting particle systems and nonlinear kinetic equations. Ann. Appl. Prob. 13, 845–889. Google Scholar
[10] Fournier, N. and Méléard, S. (2004). A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Prob. 14, 1880–1919. Google Scholar
[11] Garet, O. and Marchand, R. (2012). Asymptotic shape for the contact process in random environment. Ann. Appl. Prob. 22, 1362–1410. Google Scholar
[12] Gouéré, J.-B. and Marchand, R. (2008). Continuous first-passage percolation and continuous greedy paths model: linear growth. Ann. Appl. Prob. 18, 2300–2319. Google Scholar
[13] Howard, C. D. and Newman, C. M. (1997). Euclidean models of first-passage percolation. Prob. Theory Relat. Fields 108, 153–170. Google Scholar
[14] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York. Google Scholar
[15] Kesten, H. (1987). Percolation theory and first-passage percolation. Ann. Prob. 15, 1231–1271. Google Scholar
[16] Kondratiev, Y. G. and Kutoviy, O. V. (2006). On the metrical properties of the configuration space. Math. Nachr. 279, 774–783. Google Scholar
[17] Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Prob. 13, 1279–1285. CrossRefGoogle Scholar
[18] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin. Google Scholar
[19] Massoulié, L. (1998). Stability results for a general class of interacting point processes dynamics, and applications. Stoch. Process. Appl. 75, 1–30. Google Scholar
[20] Møller, J. and Waagepetersen, R. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, FL. Google Scholar
[21] Richardson, D. (1973). Random growth in a tessellation. Proc. Camb. Philos. Soc. 74, 515–528. Google Scholar
[22] Röckner, M. and Schied, A. (1999). Rademacher's theorem on configuration spaces and applications. J. Funct. Anal. 169, 325–356. Google Scholar
[23] Shi, Z. (2015). Branching Random Walks (Lecture Notes Math. 2151) Springer, Cham. CrossRefGoogle Scholar
[24] Tartarini, D. and Mele, E. (2016). Adult stem cell therapies for wound healing: biomaterials and computational models. Frontiers Bioeng. Biotech. 3, 10.3389/fbioe.2015.00206. Google Scholar
[25] Treloar, K. et al. (2013). Multiple types of data are required to identify the mechanisms influencing the spatial expansion of melanoma cell colonies. BMC Systems Biol. 7, 137. Google Scholar
[26] Vo, B. N., Drovandi, C. C., Pettitt, A. N. and Pettet, G. J. (2015). Melanoma cell colony expansion parameters revealed by approximate Bayesian computation. PLOS Comput. Biol. 11, e1004635. Google Scholar
[27] Waclaw, B. et al. (2015). A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity. Nature 525, 261–264. Google Scholar