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, p. 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 : https://doi.org/10.1214/13-AIHP573
Classification:  60K37,  60J80,  60G50
Keywords: random walk in random environment, Galton–Watson tree, speed, stochastic domination
@article{AIHPB_2015__51_1_304_0,
     author = {Mehrdad, Behzad and Sen, Sanchayan and Zhu, Lingjiong},
     title = {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},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {1},
     year = {2015},
     pages = {304-318},
     doi = {10.1214/13-AIHP573},
     zbl = {06412906},
     mrnumber = {3300972},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_1_304_0}
}
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://www.numdam.org/item/AIHPB_2015__51_1_304_0/

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