Transience/recurrence and the speed of a one-dimensional random walk in a “have your cookie and eat it” environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, p. 949-964

Consider a variant of the simple random walk on the integers, with the following transition mechanism. At each site x, the probability of jumping to the right is ω(x)∈[½, 1), until the first time the process jumps to the left from site x, from which time onward the probability of jumping to the right is ½. We investigate the transience/recurrence properties of this process in both deterministic and stationary, ergodic environments {ω(x)}xZ. In deterministic environments, we also study the speed of the process.

Considérons une variante de la marche aléatoire simple et symétrique sur les entiers, avec le mécanisme de transition suivant: A chaque site x, la probabilité de sauter à droite est ω(x)∈[½, 1), jusqu'à la première fois que le processus saute à gauche du site x, après laquelle la probabilité de sauter à droite est ½. Nous examinons les propriétés de transience/récurrence pour ce processus, dans les environnements déterministes et aussi dans les environnements stationnaires et ergodiques {ω(x)}xZ. Dans les environnements déterministes, nous étudions aussi la vitesse du processus.

DOI : https://doi.org/10.1214/09-AIHP331
Classification:  60K35,  60K37
Keywords: excited random walk, cookies, transience, recurrence, ballistic
@article{AIHPB_2010__46_4_949_0,
     author = {Pinsky, Ross},
     title = {Transience/recurrence and the speed of a one-dimensional random walk in a ``have your cookie and eat it'' environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     pages = {949-964},
     doi = {10.1214/09-AIHP331},
     zbl = {1218.60089},
     mrnumber = {2744879},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_4_949_0}
}
Pinsky, Ross G. Transience/recurrence and the speed of a one-dimensional random walk in a “have your cookie and eat it” environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, pp. 949-964. doi : 10.1214/09-AIHP331. http://www.numdam.org/item/AIHPB_2010__46_4_949_0/

[1] A. Basdevant and A. Singh. On the speed of a cookie random walk. Probab. Theory Related Fields 141 (2008) 625-645. | MR 2391167 | Zbl 1141.60383

[2] I. Benjamini and D. Wilson. Excited random walk. Electron. Comm. Probab. 8 (2003) 86-92. | MR 1987097 | Zbl 1060.60043

[3] B. Davis. Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113 (1999) 501-518. | MR 1717528 | Zbl 0930.60041

[4] R. Durrett. Probability: Theory and Examples, 3rd edition. Brooks/Cole-Thomson Learning, Belmont, CA, 2005. | MR 1068527 | Zbl 0709.60002

[5] E. Kosygina and M. Zerner. Positively and negatively excited random walks on intergers, with branching processes. Electron. J. Probab. 13 (2008) 1952-1979. | MR 2453552 | Zbl 1191.60113

[6] M. Zerner. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005) 98-122. | MR 2197139 | Zbl 1076.60088