Consider biased random walks on two Galton–Watson trees without leaves having progeny distributions and ( and ) where and are supported on positive integers and dominates stochastically. We prove that the speed of the walk on is bigger than the same on when the bias is larger than a threshold depending on and . 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 et ayant des lois de reproduction respectivement et , deux lois supportées par les entiers positifs telles que domine stochastiquement . Nous prouvons que la vitesse de la marche sur est supérieure ou égale á celle sur si le biais est plus grand qu’un seuil dépendant de et . Ceci répond partiellement á une question posée par Ben Arous, Fribergh et Sidoravicius (Comm. Pure Appl. Math. 67 (2014) 519–530).
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{\textendash}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}, pages = {304--318}, publisher = {Gauthier-Villars}, volume = {51}, number = {1}, year = {2015}, doi = {10.1214/13-AIHP573}, mrnumber = {3300972}, zbl = {1314.60160}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/13-AIHP573/} }
TY - JOUR AU - Mehrdad, Behzad AU - Sen, Sanchayan AU - Zhu, Lingjiong TI - 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 JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 304 EP - 318 VL - 51 IS - 1 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/13-AIHP573/ DO - 10.1214/13-AIHP573 LA - en ID - AIHPB_2015__51_1_304_0 ER -
%0 Journal Article %A Mehrdad, Behzad %A Sen, Sanchayan %A Zhu, Lingjiong %T 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 %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 304-318 %V 51 %N 1 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/13-AIHP573/ %R 10.1214/13-AIHP573 %G en %F 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://archive.numdam.org/articles/10.1214/13-AIHP573/
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