The spread of a catalytic branching random walk

Carmona, Philippe; Hu, Yueyun

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 $\mathbb{Z}$ 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 $\alpha$ explicite telle que $\frac{M_n}{n}\to \alpha$ 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 $n\to +\infty$, pour la suite $M_n-\alpha n$.

We consider a catalytic branching random walk on $\mathbb{Z}$ 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 $\alpha$, $\frac{M_n}{n}\to \alpha$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_n-\alpha n$ as $n$ goes to infinity.

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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.

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl