The spread of a catalytic branching random walk
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 327-351.

Nous considérons une marche aléatoire branchant catalytique sur qui ne branche qu’à l’origine. Dans le cas surcritique, nous établissons une loi des grands nombres pour la position maximale M n : Il existe une constante α explicite telle que M n nα presque sûrement sur l’ensemble des trajectoires pour lesquelles l’origine est visitée une infinité de fois. Ensuite, nous déterminons toutes les lois limites possibles, lorsque n+, pour la suite M n -αn.

We consider a catalytic branching random walk on that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position M n : For some constant α, M n nα almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for M n -αn as n goes to infinity.

DOI : https://doi.org/10.1214/12-AIHP529
Classification : 60K37
Mots clés : branching processes, catalytic branching random walk
@article{AIHPB_2014__50_2_327_0,
     author = {Carmona, Philippe and Hu, Yueyun},
     title = {The spread of a catalytic branching random walk},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {327--351},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     doi = {10.1214/12-AIHP529},
     zbl = {1291.60208},
     mrnumber = {3189074},
     language = {en},
     url = {www.numdam.org/item/AIHPB_2014__50_2_327_0/}
}
Carmona, Philippe; Hu, Yueyun. The spread of a catalytic branching random walk. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 327-351. doi : 10.1214/12-AIHP529. http://www.numdam.org/item/AIHPB_2014__50_2_327_0/

[1] E. Aïdékon. Convergence in law of the minimum of a branching random walk. Preprint. Ann. Probab. To appear. Available at http://arxiv.org/abs/1101.1810. | MR 3098680 | Zbl 1285.60086

[2] E. Aïdékon and Z. Shi. Weak convergence for the minimal position in a branching random walk: A simple proof. Period. Math. Hungar. 61(1-2) (2010) 43-54. | MR 2728431 | Zbl 1240.60227

[3] S. Albeverio and L. V. Bogachev. Branching random walk in a catalytic medium. I. Basic equations. Positivity 4(1) (2000) 41-100. | MR 1740207 | Zbl 0953.60079

[4] S. Albeverio, L. V. Bogachev and 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. | MR 1649878 | Zbl 0917.60080

[5] S. Albeverio, L. V. Bogachev and E. B. Yarovaya. Erratum: “Asymptotics of branching symmetric random walk on the lattice with a single source”. C. R. Acad. Sci. Paris Sér. I Math. 327(6) (1998) 585. | MR 1650599 | Zbl 0917.60080

[6] K. B. Athreya and P. E. Ney. Branching Processes. Dover, Mineola, NY, 2004. Reprint of the 1972 original [Springer, New York; MR0373040]. | MR 373040 | Zbl 1070.60001

[7] J. Berestycki, É. Brunet, J. W. Harris and S. C. Harris. The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential. Statist. Probab. Lett. 80(17-18) (2010) 1442-1446. | MR 2669786 | Zbl 1196.60144

[8] J. D. Biggins. Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1977) 25-37. | MR 433619 | Zbl 0356.60053

[9] J. D. Biggins and A. E. Kyprianou. Measure change in multitype branching. Adv. in Appl. Probab. 36(2) (2004) 544-581. | MR 2058149 | Zbl 1056.60082

[10] L. V. Bogachev and E. B. Yarovaya. Moment analysis of a branching random walk on a lattice with a single source. Dokl. Akad. Nauk 363(4) (1998) 439-442. | MR 1702745 | Zbl 0963.60083

[11] M. Bramson. Minimal displacement of branching random walk. Z. Wahrsch. Verw. Gebiete 45 (1978) 89-108. | MR 510529 | Zbl 0373.60089

[12] Ph. Carmona. A large deviation theory via the renewal theorem. Note, 2005. Available at http://www.math.sciences.univ-nantes.fr/~carmona/renewaldp.pdf.

[13] K. S. Crump. On systems of renewal equations. J. Math. Anal. Appl. 30 (1970) 425-434. | MR 257678 | Zbl 0198.22502

[14] B. De Saporta. Renewal theorem for a system of renewal equations. Ann. Inst. Henri Poincaré Probab. Stat. 39 (2003) 823-838. | Numdam | MR 1997214 | Zbl 1021.60069

[15] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Stoch. Model. Appl. Probab. 38. Springer, Berlin, 2010. Corrected reprint of the second (1998) edition. | MR 2571413 | Zbl 1177.60035

[16] L. Döring and M. Roberts. Catalytic branching processes via spine techniques and renewal theory. Preprint, 2011. Available at http://arxiv.org/abs/1106.5428. | Zbl pre06234279

[17] L. Döring and M. Savov. An application of renewal theorems to exponential moments of local times. Electron. Commun. Probab. 15 (2010), 263-269. | MR 2658973 | Zbl 1226.60103

[18] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. I. Wiley, New York, 1950. | MR 38583 | Zbl 0039.13201

[19] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York, 1966. | MR 210154 | Zbl 0219.60003

[20] R. Hardy and S. C. Harris. A spine approach to branching diffusions with applications to p -convergence of martingales. In Séminaire de Probabilités XLII 281-330, 2009. | MR 2599214 | Zbl 1193.60100

[21] J. W. Harris and S. C. Harris. Branching Brownian motion with an inhomogeneous breeding potential. Ann. Inst. Henri Poincaré Probab. Stat. 45(3) (2009) 793-801. | Numdam | MR 2548504 | Zbl 1183.60029

[22] S. C. Harris and M. I. Roberts. The many-to-few lemma and multiple spines. Preprint, 2011. Available at http://arxiv.org/abs/1106.4761.

[23] Y. Hu and Z. Shi. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37(2) (2009) 742-789. | MR 2510023 | Zbl 1169.60021

[24] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of LlogL criteria for mean behavior of branching processes. Ann. Probab. 23(3) (1995) 1125-1138. | MR 1349164 | Zbl 0840.60077

[25] P. Révész. Random Walks of Infinitely Many Particles. World Scientific, River Edge, NJ, 1994. | MR 1645302 | Zbl 0841.60053

[26] Z. Shi. Branching random walks. Saint-Flour summer's course, 2012.

[27] V. Topchii and V. Vatutin. Individuals at the origin in the critical catalytic branching random walk. In Discrete Random Walks (Paris, 2003) 325-332 (electronic). Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2003. | MR 2042398 | Zbl 1034.60078

[28] V. Topchii and V. Vatutin. Two-dimensional limit theorem for a critical catalytic branching random walk. In Mathematics and Computer Science III 387-395. Birkhäuser, Basel, 2004. | MR 2090528 | Zbl 1062.60091

[29] V. A. Vatutin and V. A. Topchiĭ. A limit theorem for critical catalytic branching random walks. Teor. Veroyatn. Primen. 49(3) (2004), 461-484. | MR 2144864 | Zbl 1093.60062

[30] V. A. Vatutin, V. A. Topchiĭ and E. B. Yarovaya. Catalytic branching random walks and queueing systems with a random number of independently operating servers. Teor. Ĭmovīr. Mat. Stat. 69 (2003) 1-15. | MR 2110900 | Zbl 1097.60068