The speed of a biased walk on a Galton–Watson tree without leaves is monotonic with respect to progeny distributions for high values of bias
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, pp. 304-318.

Consider biased random walks on two Galton–Watson trees without leaves having progeny distributions P 1 and P 2 ( GW (P 1 ) and GW (P 2 )) where P 1 and P 2 are supported on positive integers and P 1 dominates P 2 stochastically. We prove that the speed of the walk on GW (P 1 ) is bigger than the same on GW (P 2 ) when the bias is larger than a threshold depending on P 1 and P 2 . This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius (Comm. Pure Appl. Math. 67 (2014) 519–530).

Nous considérons des marches aléatoires biaisées sur deux arbres de Galton–Watson sans feuilles GW (P 1 ) et GW (P 2 ) ayant des lois de reproduction respectivement P 1 et P 2 , deux lois supportées par les entiers positifs telles que P 1 domine stochastiquement P 2 . Nous prouvons que la vitesse de la marche sur GW (P 1 ) est supérieure ou égale á celle sur GW (P 2 ) si le biais est plus grand qu’un seuil dépendant de P 1 et P 2 . Ceci répond partiellement á une question posée par Ben Arous, Fribergh et Sidoravicius (Comm. Pure Appl. Math. 67 (2014) 519–530).

DOI: 10.1214/13-AIHP573
Classification: 60K37, 60J80, 60G50
Keywords: random walk in random environment, Galton–Watson tree, speed, stochastic domination
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Mehrdad, Behzad; Sen, Sanchayan; Zhu, Lingjiong. The speed of a biased walk on a Galton–Watson tree without leaves is monotonic with respect to progeny distributions for high values of bias. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, pp. 304-318. doi : 10.1214/13-AIHP573. http://archive.numdam.org/articles/10.1214/13-AIHP573/

[1] E. Aïdékon. Speed of the biased random walk on a Galton–Watson tree. Probab. Theory Related Fields. To appear, 2014. DOI:10.1007/s00440-013-0515-y. | DOI | MR | Zbl

[2] K. B. Athreya. Large deviation rates for branching processes. I. Single type case. Ann. Appl. Probab. 4 (1994) 779–790. | MR | Zbl

[3] G. Ben Arous, A. Fribergh and V. Sidoravicius. Lyons–Pemantle–Peres monotonicity problem for high biases. Comm Pure Appl. Math. 67 (2014) 519–530. | DOI | MR | Zbl

[4] N. H. Bingham. On the limit of a supercritical branching process. J. Appl. Probab. 25 (1988) 215–228. | DOI | MR | Zbl

[5] H. Kesten and B. P. Stigum. A limit theorem for multidimensional Galton–Watson processes. Ann. Math. Statist. 37 (1966) 1211–1223. | DOI | MR | Zbl

[6] R. Lyons. Random walks and percolation on trees. Ann. Probab. 18 (1990) 931–958. | DOI | MR | Zbl

[7] R. Lyons, R. Pemantle and Y. Peres. Ergodic theory on Galton–Watson trees: Speed of random walk and dimension of harmonic measure. Erg. Theory Dynam. Syst. 15 (1995) 593–619. | DOI | MR | Zbl

[8] R. Lyons, R. Pemantle and Y. Peres. Biased random walks on Galton–Watson trees. Probab. Theory Related Fields 106 (1996) 254–268. | DOI | MR | Zbl

[9] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of LlogL criteria for mean behavior of branching processes. Ann. Probab. 23 (1995) 1125–1138. | DOI | MR | Zbl

[10] O. Zeitouni. Random walks in random environment. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837 189–312. Springer, Berlin, 2004. | MR | Zbl

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