Geodesic
Simulation of branching random walks on a~multidimensional lattice
Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2018) no. 3, pp. 37-56.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider continuous-time branching random walks on multidimensional lattices with birth and death of particles at a finite number of lattice points. Such processes are used in numerous applications, in particular, in statistical physics, population dynamics, and chemical kinetics. In the last decade, for various models of branching random walks, a series of limit theorems about the behavior of the process for large times has been obtained. However, it is almost impossible to analyze analytically branching random walks on finite time intervals; so in this paper we present an algorithm for simulating branching random walks and examples of its numerical realization. 
@article{FPM_2018_22_3_a2,
     author = {E. M. Ermishkina and E. B. Yarovaya},
     title = {Simulation of branching random walks on a~multidimensional lattice},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {37--56},
     publisher = {mathdoc},
     volume = {22},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2018_22_3_a2/}
}
TY  - JOUR
AU  - E. M. Ermishkina
AU  - E. B. Yarovaya
TI  - Simulation of branching random walks on a~multidimensional lattice
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2018
SP  - 37
EP  - 56
VL  - 22
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2018_22_3_a2/
LA  - ru
ID  - FPM_2018_22_3_a2
ER  - 
%0 Journal Article
%A E. M. Ermishkina
%A E. B. Yarovaya
%T Simulation of branching random walks on a~multidimensional lattice
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2018
%P 37-56
%V 22
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2018_22_3_a2/
%G ru
%F FPM_2018_22_3_a2
E. M. Ermishkina; E. B. Yarovaya. Simulation of branching random walks on a~multidimensional lattice. Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2018) no. 3, pp. 37-56. http://geodesic.mathdoc.fr/item/FPM_2018_22_3_a2/

[1] Daletskii Yu. L., Krein M. G., Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970

[2] Zenyuk D. A., “Modelirovanie sluchainykh bluzhdanii na regulyarnykh fraktalnykh mnozhestvakh”, Matem. modelirovanie, 26:11 (2014), 101–104 | MR | Zbl

[3] Molchanov S. A., Yarovaya E. B., “Struktura populyatsii vnutri rasprostranyayuschegosya fronta vetvyaschegosya sluchainogo bluzhdaniya s konechnym chislom tsentrov generatsii chastits”, Dokl. RAN, 447:3 (2012), 265–268 | Zbl

[4] Yarovaya E. B., Vetvyaschiesya sluchainye bluzhdaniya v neodnorodnoi srede, Tsentr prikl. issled. pri mekhaniko-matematicheskom f-te MGU, M., 2007

[5] Yarovaya E. B., “Kriterii eksponentsialnogo rosta chisla chastits v modelyakh vetvyaschikhsya sluchainykh bluzhdanii”, Teoriya veroyatn. i ee primen., 55:4 (2010), 705–731 | DOI

[6] Yarovaya E. B., “Spektralnye svoistva evolyutsionnykh operatorov v modelyakh vetvyaschikhsya bluzhdanii s neskolkimi istochnikami vetvleniya”, Matem. zametki, 92:1 (2012), 124–140 | DOI

[7] Yarovaya E. B., “Struktura polozhitelnogo diskretnogo spektra evolyutsionnogo operatora vetvyaschikhsya sluchainykh bluzhdanii”, Dokl. RAN, 463:6 (2015), 646–649 | DOI | Zbl

[8] Albeverio S., Bogachev L. V., Yarovaya E. B., “Asymptotics of branching symmetric random walk on the lattice with a single source”, C. R. Acad. Sci. Paris Sér. I Math., 326:8 (1998), 975–980 | DOI | MR | Zbl

[9] Andries E., Umarov S., Steinberg S., “Monte Carlo random walk simulations based on distributed order differential equations with applications to cell biology”, Fract. Calc. Appl. Anal., 9:4 (2006), 351–369 | MR | Zbl

[10] Bramson M., Ney P., Tao J., “The population composition of a multitype branching random walk”, Ann. Appl. Probab., 2:3 (1992), 575–596 | DOI | MR | Zbl

[11] Cormen T. H., Leiserson C. E., Rivest R. L., Stein C., Introduction to Algorithms, MIT Press, Cambridge, 2009 | MR | Zbl

[12] Daskalova N., “EM algorithm for estimation of the offspring probabilities in some branching models”, Nonlinear Dynamics of Electronic Systems, eds. V. M. Mladenov, P. Ch. Ivanov, Springer, Cham, 2014, 181–188 | DOI | Zbl

[13] J. Gärtner, S. A. Molchanov, “"Parabolic problems for the Anderson model. I. Intermittency and related topics”, Comm. Math. Phys., 132:3 (1990), 613–655 | DOI | MR

[14] Ho L. S. T., Xu J., Crawford F. W., Minin V. N., Suchard M. A., “Birth/birth-death processes and their computable transition probabilities with biological applications”, J. Math. Biology, 76:4 (2018), 911–944 | DOI | MR | Zbl

[15] Kleinhans D., Friedrich R., “Maximum likelihood estimation of drift and diffusion functions”, Phys. Lett. A, 368:3–4 (2007), 194–198 | DOI | MR | Zbl

[16] Kolomeisky A. B., Fisher M. E., “Periodic sequential kinetic models with jumping, branching and deaths”, Phys. A, 279:1–4 (2000), 1–20 | DOI | MR

[17] Molchanov S., “Reaction-diffusion equations in the random media: localization and intermittency”, Nonlinear Stochastic PDEs, IMA Vol. Math. Its Appl., 77, eds. Funaki T., Woyczynski W. A., Springer, New York, 1996, 81–109 | DOI | MR | Zbl

[18] Moral P. D., Jasra A., Note on random walks with absorbing barriers and sequential Monte Carlo methods, 2016, arXiv: 1611.03177 | MR

[19] Murai F., Ribeiro B., Towsley D., Gile K., “Characterizing branching processes from sampled data”, WWW '13 Companion Proc. of the 22nd Int. Conf. on World Wide Web, ACM, New York, 2013, 805–812

[20] Tuerlinckx F., Maris E., Ratcliff R., De Boeck P., “A comparison of four methods for simulating the diffusion process”, Behavior Res. Methods, Instruments, Computers, 33:4 (2001), 443–456 | DOI

[21] Vatutin V. A., Topchiĭ V. A., Yarovaya E. B., “Catalytic branching random walks and queueing systems with a random number of independent servers”, Teor. \u Imovir. Mat. Stat., 69 (2003), 1–15 | MR | Zbl

[22] Yarovaya E. B., “Branching random walks with several sources”, Math. Popul. Stud., 20:1 (2013), 14–26 | DOI | MR | Zbl

[23] Yarovaya E. B., “Positive discrete spectrum of the evolutionary operator of supercritical branching walks with heavy tails”, Methodology Comput. Appl. Probab., 19:4 (2016), 1151–1167 | DOI | MR

[24] Zeldovich Ya. B., Molchanov S. A., Ruzmaikin A. A., Sokoloff D. D., Intermittency, Diffusion and Generation in a Nonstationary Random Medium, v. 1, Rev. Math. Math. Phys., 15, Cambridge Sci. Publ., Cambridge, 2014 | MR