Fluctuations of the propagation front of a catalytic branching walk

E. Vl. Bulinskaya

Teoriâ veroâtnostej i ee primeneniâ, Tome 64 (2019) no. 4, pp. 642-670

Résumé

We consider a supercritical catalytic branching random walk (CBRW) with finite number of catalysts on a multidimensional lattice $\mathbb{Z}^d$, $d\in\mathbf{N}$. The behavior of a cloud of particles in space and time is studied. When estimating the rate of the population propagation for the front of a multidimensional CBRW, Bulinskaya [Stochastic Process. Appl., 128 (2018), pp. 2325–2340] extended the strong limit theorem by Carmona and Hu [Ann. Inst. Henri Poincaré Probab. Stat., 50 (2014), pp. 327–351]. Our aim is to analyze the fluctuations of the propagation front of a CBRW on $\mathbb{Z}^d$.

Détail

Keywords: catalytic branching random walk, supercritical regime, spread of population
Mots-clés : propagation front, fluctuations of front.

Bibliographie

[1] J. D. Biggins, “The asymptotic shape of the branching random walk”, Adv. in Appl. Probab., 10:1 (1978), 62–84 | DOI | MR | Zbl

[2] Zhan Shi, Branching random walks. École d'Été de Probabilités de Saint-Flour XLII – 2012, Lecture Notes in Math., 2151, Springer, Cham, 2015, x+133 pp. | DOI | MR | Zbl

[3] B. Mallein, “Asymptotic of the maximal displacement in a branching random walk”, Grad. J. Math., 1:2 (2016), 92–104 | MR

[4] A. Bhattacharya, R. S. Hazra, P. Roy, “Branching random walks, stable point processes and regular variation”, Stochastic Process. Appl., 128:1 (2018), 182–210 | DOI | MR | Zbl

[5] S. Albeverio, L. V. Bogachev, E. B. Yarovaya, “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

[6] V. A. Vatutin, V. A. Topchii, E. B. Yarovaya, “Catalytic branching random walk and queueing systems with random number of independent servers”, Theory Probab. Math. Statist., 2004, no. 69, 1–15 | DOI | MR | Zbl

[7] Y. Hu, V. A. Topchii, V. A. Vatutin, “Branching random walk in $\mathbf Z^4$ with branching at the origin only”, Theory Probab. Appl., 56:2 (2011), 193–212 | DOI | DOI | MR | Zbl

[8] S. A. Molchanov, E. B. Yarovaya, “Branching processes with lattice spatial dynamics and a finite set of particle generation centers”, Dokl. Math., 86:2 (2012), 638–641 | DOI | MR | Zbl

[9] Ph. Carmona, Yueyun Hu, “The spread of a catalytic branching random walk”, Ann. Inst. Henri Poincaré Probab. Stat., 50:2 (2014), 327–351 | DOI | MR | Zbl

[10] E. Vl. Bulinskaya, “Maximum of a catalytic branching random walk”, Russian Math. Surveys, 74:3 (2019), 546–548 | DOI | DOI | MR

[11] E. Vl. Bulinskaya, “Spread of a catalytic branching random walk on a multidimensional lattice”, Stochastic Process. Appl., 128:7 (2018), 2325–2340 | DOI | MR | Zbl

[12] E. Vl. Bulinskaya, Maximum of catalytic branching random walk with regularly varying tails, 2018, arXiv: 1808.01465

[13] E. Vl. Bulinskaya, Catalytic branching random walk with semi-exponential increments, 2019, arXiv: 1902.00994

[14] E. Vl. Bulinskaya, “Multidimensional catalytic branching random walk with regularly varying tails”, ICoMS'19 Proceedings of the 2019 2nd international conference on mathematics and statistics (Prague, 2019), ACM, New York, 2019, 6–13 | DOI

[15] E. Vl. Bulinskaya, “Complete classification of catalytic branching processes”, Theory Probab. Appl., 59:4 (2015), 545–566 | DOI | DOI | MR | Zbl

[16] E. Vl. Bulinskaya, “Strong and weak convergence of the population size in a supercritical catalytic branching process”, Dokl. Math., 92:3 (2015), 714–718 | DOI | MR | Zbl

[17] M. V. Platonova, K. S. Ryadovkin, “On the mean number of particles of a branching random walk on $\mathbb{Z}^d$ with periodic sources of branching”, Dokl. Math., 97:2 (2018), 140–143 | DOI | MR | Zbl

[18] B. A. Sewastjanow, Verzweigungsprozesse, Math. Lehrbücher Monogr. II. Abt. Math. Monogr., 34, Akademie-Verlag, Berlin, 1974, xi+326 pp. | MR | MR | Zbl | Zbl

[19] V. A. Vatutin, “Vetvyaschiesya protsessy Bellmana–Kharrisa”, Lekts. kursy NOTs, 12, MIAN, M., 2009, 3–111 | DOI | Zbl

[20] E. Vl. Bulinskaya, “Finiteness of hitting times under taboo”, Statist. Probab. Lett., 85 (2014), 15–19 | DOI | MR | Zbl

[21] P. Brémaud, Markov chains. Gibbs fields, Monte-Carlo simulation, and queues, Texts Appl. Math., 31, Springer-Verlag, New York, 1999, xviii+444 pp. | DOI | MR | Zbl

[22] R. T. Rockafellar, Convex analysis, Princeton Math. Ser., 28, Princeton Univ. Press, Princeton, NJ, 1970, xviii+451 pp. | MR | Zbl | Zbl

[23] A. A. Borovkov, Probability theory, Universitext, Springer, London, 2013, xxviii+733 pp. | DOI | MR | Zbl

[24] A. A. Borovkov, K. A. Borovkov, Asymptotic analysis of random walks. Heavy-tailed distributions, Encyclopedia Math. Appl., Cambridge Univ. Press, Cambridge, 118, xxx+625 pp. | DOI | MR | Zbl

[25] A. A. Borovkov, Asimptoticheskii analiz sluchainykh bluzhdanii. Bystro ubyvayuschie raspredeleniya priraschenii, Fizmatlit, M., 2013, 447 pp. | Zbl

[26] D. R. Cox, Renewal theory, Methuen Co. Ltd., London; John Wiley Sons, Inc., New York, 1962, ix+142 pp. | MR | MR | Zbl | Zbl

[27] M. V. Fedoryuk, Asimptotika: integraly i ryady, Nauka, M., 1987, 544 pp. | MR | Zbl

[28] K. S. Crump, “On systems of renewal equations”, J. Math. Anal. Appl., 30:2 (1970), 425–434 | DOI | MR | Zbl

[29] N. Kaplan, “The supercritical multitype age-dependent branching process”, J. Math. Anal. Appl., 50 (1975), 164–182 | DOI | MR | Zbl

[30] K. B. Athreya, “On the supercritical one dimensional age dependent branching processes”, Ann. Math. Stat., 40:3 (1969), 743–763 | DOI | MR | Zbl