1. Home
  2. Issues
  3. Volume 9, Issue 3 (2022)
  4. Spatial birth-and-death processes with a ...

Modern Stochastics: Theory and Applications


Spatial birth-and-death processes with a finite number of particles
Volume 9, Issue 3 (2022), pp. 279–312
Pub. online: 19 April 2022      Type: Research Article      Open accessOpen Access
Received
27 September 2021
Revised
14 March 2022
Accepted
14 March 2022
Published
19 April 2022

Abstract

The aim of this work is to establish essential properties of spatial birth-and-death processes with general birth and death rates on ${\mathbb{R}^{\mathrm{d}}}$. Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over ${\mathbb{R}^{\mathrm{d}}}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator.
The pathwise behavior of an aggregation model is also studied. The probability of extinction and the growth rate of the number of particles under condition of nonextinction are estimated.

References

[1] 
Athreya, K.B., Ney, P.E.: Branching Processes. Springer (1972). Die Grundlehren der mathematischen Wissenschaften
[2] 
Baccelli, F., Mathieu, F., Norros, I.: Mutual service processes in Euclidean spaces: existence and ergodicity. Queueing Syst. 86(1-2), 95–140 (2017). MR3642012. https://doi.org/10.1007/s11134-017-9524-3
[3] 
Barczy, M., Li, Z., Pap, G.: Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration. ALEA, Lat. Am. J. Probab. Math. Stat. 12(1), 129–169 (2015) MR3340375
[4] 
Belavkin, V.P., Kolokoltsov, V.N.: On a general kinetic equation for many-particle systems with interaction, fragmentation and coagulation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459(2031), 727–748 (2003) MR1996729. https://doi.org/10.1098/rspa.2002.1026
[5] 
Bezborodov, V.: Corrigendum to: Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes. Adv. Appl. Probab. 52(4), 1325–1327 (2020) MR4190043. https://doi.org/10.1017/apr.2020.52
[6] 
Bezborodov, V., di Persio, L., Finkelshtein, D., Kondratiev, Y., Kutoviy, O.: Fecundity regulation in a spatial birth-and-death process. Stoch. Dyn. 21(1), 27 (2021). MR4192900. https://doi.org/10.1142/S0219493720500380
[7] 
Bezborodov, V., Di Persio, L., Krueger, T., Lebid, M., Ożański, T.: Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes. Adv. Appl. Probab. 50(1), 74–101 (2017) MR3781978. https://doi.org/10.1017/apr.2018.5
[8] 
Bezborodov, V., Persio, L.D.: Maximal irreducibility measure for spatial birth-and-death processes. Stat. Probab. Lett. 125, 25–32 (2017) MR3626065. https://doi.org/10.1016/j.spl.2017.01.019
[9] 
Bezborodov, V., Kondratiev, Y., Kutoviy, O.: Lattice birth-and-death processes. Mosc. Math. J. 19(1), 7–36 (2019) MR3934306. https://doi.org/10.17323/1609-4514-2019-19-1-7-36
[10] 
Bolker, B.M., Pacala, S.W.: Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol. 52, 179–197 (1997)
[11] 
Bolker, B.M., Pacala, S.W.: Spatial moment equations for plant competitions: Understanding spatial strategies and the advantages of short dispersal. Am. Nat. 153(6), 575–602 (1999)
[12] 
Bulluck, L., Fleishman, E., Betrus, C., Blair, R.: Spatial and temporal variations in species occurrence rate affect the accuracy of occurrence models. Global Ecology and Biogeography 15(1), 27–38 (2006)
[13] 
Burke, C.J., Rosenblatt, M.: A Markovian function of a Markov chain. Ann. Math. Stat. 29, 1112–1122 (1958) MR0101557. https://doi.org/10.1214/aoms/1177706444
[14] 
Chung, K.L.: Markov Chains with Stationary Transition Probabilities. Springer (1967) MR0217872
[15] 
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Vol. II, 2nd edn. Probability and its Applications, p. 573. Springer (2008). General theory and structure MR2371524. https://doi.org/10.1007/978-0-387-49835-5
[16] 
Durrett, R.: Lecture Notes on Particle Systems and Percolation. The Wadsworth & Brooks/Cole Statistics/Probability Series, p. 335. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA (1988) MR0940469
[17] 
Eibeck, A., Wagner, W.: Stochastic interacting particle systems and nonlinear kinetic equations. Ann. Appl. Probab. 13(3), 845–889 (2003) MR1994039. https://doi.org/10.1214/aoap/1060202829
[18] 
Etheridge, A.M.: Survival and extinction in a locally regulated population. Ann. Appl. Probab. 14(1), 188–214 (2004) MR2023020. https://doi.org/10.1214/aoap/1075828051
[19] 
Etheridge, A.M., Kurtz, T.G.: Genealogical constructions of population models. Ann. Probab. 47(4), 1827–1910 (2019). MR3980910. https://doi.org/10.1214/18-AOP1266
[20] 
Finkelshtein, D., Kondratiev, Y., Kutoviy, O.: Semigroup approach to birth-and-death stochastic dynamics in continuum. J. Funct. Anal. 262(3), 1274–1308 (2012) MR2863863. https://doi.org/10.1016/j.jfa.2011.11.005
[21] 
Finkelshtein, D., Kondratiev, Y., Kutoviy, O., Zhizhina, E.: On an aggregation in birth-and-death stochastic dynamics. Nonlinearity 27(6), 1105–1133 (2014) MR3207926. https://doi.org/10.1088/0951-7715/27/6/1105
[22] 
Finkelshtein, D., Kondratiev, Y., Kozitsky, Y., Kutoviy, O.: The statistical dynamics of a spatial logistic model and the related kinetic equation. Math. Models Methods Appl. Sci. 25(2), 343–370 (2015). MR3280962. https://doi.org/10.1142/S0218202515500128
[23] 
Fournier, N., Méléard, S.: A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14(4), 1880–1919 (2004) MR2099656. https://doi.org/10.1214/105051604000000882
[24] 
Garcia, N.L., Kurtz, T.G.: Spatial birth and death processes as solutions of stochastic equations. ALEA Lat. Am. J. Probab. Math. Stat. 1, 281–303 (2006) MR2249658
[25] 
Garcia, N.L., Kurtz, T.G.: Spatial point processes and the projection method. In: In and Out of Equilibrium. 2. Progr. Probab., vol. 60, pp. 271–298. Birkhäuser, Basel (2008) MR2477386. https://doi.org/10.1007/978-3-7643-8786-0_13
[26] 
Harris, T.E.: The Theory of Branching Processes. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen. Springer (1963)
[27] 
Holley, R.A., Stroock, D.W.: Nearest neighbor birth and death processes on the real line. Acta Math. 140(1-2), 103–154 (1978) MR0488380. https://doi.org/10.1007/BF02392306
[28] 
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland Mathematical Library, vol. 24, p. 555. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo (1989) MR1011252
[29] 
Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Probability and its Applications. Springer (2002) MR1876169. https://doi.org/10.1007/978-1-4757-4015-8
[30] 
Kondratiev, Y., Kuna, T.: Harmonic analysis on configuration space. I. General theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(2), 201–233 (2002) MR1914839. https://doi.org/10.1142/S0219025702000833
[31] 
Lavancier, F., Le Guével, R.: Spatial birth–death–move processes: Basic properties and estimation of their intensity functions. J. R. Stat. Soc., Ser. B, Stat. Methodol. 83(4), 798–825 (2021) MR4320002. https://doi.org/10.1111/rssb.12452
[32] 
Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes vol. 324. Springer (1999) MR1717346. https://doi.org/10.1007/978-3-662-03990-8
[33] 
Lopes Garcia, N.: Birth and death processes as projections of higher-dimensional Poisson processes. Adv. Appl. Probab. 27(4), 911–930 (1995) MR1358900. https://doi.org/10.2307/1427928
[34] 
Méléard, S., Tran, V.C.: Trait substitution sequence process and canonical equation for age-structured populations. J. Math. Biol. 58(6), 881–921 (2009). MR2495555. https://doi.org/10.1007/s00285-008-0202-2
[35] 
Møller, J.: On the rate of convergence of spatial birth-and-death processes. Ann. Inst. Stat. Math. 41(3), 565–581 (1989) MR1032600. https://doi.org/10.1007/BF00050669
[36] 
Møller, J., Sørensen, M.: Statistical analysis of a spatial birth-and-death process model with a view to modelling linear dune fields. Scand. J. Stat. 21(1), 1–19 (1994) MR1267040
[37] 
Ovaskainen, O., Finkelshtein, D., Kutovyi, O., Cornell, S., Bolker, B.M., Kondratiev, Y.: A mathematical framework for the analysis of spatiotemporal point processes. Theoretical Ecology 7, 101–113 (2014)
[38] 
Preston, C.: Spatial birth-and-death processes. Proc. of the 40th Session (Warsaw, 1975), Vol. 2. Invited papers. Bull. Int. Stat. Inst. 46, 2, 371–391 (1975) MR0474532
[39] 
Robinson, S.E., Christley, R.M.: Identifying temporal variation in reported births, deaths and movements of cattle in britain. BMC Veterinary Research 2(1), 11 (2006)
[40] 
Sadahiro, Y.: Analysis of the appearance and disappearance of point objects over time. Int. J. Geogr. Inf. Sci. 33(2), 215–239 (2019). https://doi.org/10.1080/13658816.2018.1538517
[41] 
Shcherbakov, V., Volkov, S.: Long term behaviour of locally interacting birth-and-death processes. J. Stat. Phys. 158(1), 132–157 (2015) MR3296278. https://doi.org/10.1007/s10955-014-1122-8

Full article PDF XML
Full article PDF XML

Copyright
© 2022 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Spatial birth-and-death processes stochastic equation graphical representation interacting particles

MSC2010
60K35 60J25

Funding
VB was supported by the Department of Computer Science at the University of Verona. VB acknowledges a partial support of the DFG through the SFB 701 (Bielefeld University) and the IRTG (IGK) 1132 “Stochastics and Real World Models”.

Metrics
since March 2018
638

Article info
views

616

Full article
views

340

PDF
downloads

121

XML
downloads


Share


RSS