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, Tome 46 (2010) no. 4, pp. 949-964.

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)}x∈Z. Dans les environnements déterministes, nous étudions aussi la vitesse du processus.

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)}x∈Z. In deterministic environments, we also study the speed of the process.

DOI : 10.1214/09-AIHP331
Classification : 60K35, 60K37
Mots-clés : excited random walk, cookies, transience, recurrence, ballistic 
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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, Tome 46 (2010) no. 4, pp. 949-964. doi : 10.1214/09-AIHP331. http://www.numdam.org/articles/10.1214/09-AIHP331/

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