Abstract
We introduce a class of nearest-neighbor integer random walks in random and non-random media, which includes excited random walks considered in the literature. At each site the random walker has a drift to the right, the strength of which depends on the environment at that site and on how often the walker has visited that site before. We give exact criteria for recurrence and transience and consider the speed of the walk.
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Benjamini, I., Wilson, D.B.: Excited random walk. Elect. Comm. Probab. 8, 86–92 (2003)
Davis, B.: Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113, 501–518 (1999)
Davis, B., Volkov, S.: Continuous time vertex-reinforced jump processes. Probab. Theory Related Fields 123, 281–300 (2002)
Durrett, R.: Probability: Theory and Examples. Pacific Grove, Calif.: Wadsworth & Brooks/Cole Advanced Books & Software, 1991
Feller, W.: An Introduction to Probability Theory and its Applications Vol. 1, 3rd ed., 1970
Kozma, G.: Excited random walk in three dimensions has positive speed. Preprint, 2003
Perman, M., Werner, W.: Perturbed Brownian motions. Probab. Theory Related Fields 108, 357–383 (1997)
Solomon, F.: Random walks in a random environment. Ann. Probab. 3, 1–31 (1975)
Sznitman, A.-S.: Topics in Random Walks in Random Environment. School and Conference on Probability Theory, ICTP Lecture Notes Series, Trieste, 203–266 (2004)
Sznitman, A.-S., Zerner, M.P.W.: A law of large numbers for random walks in random environment. Ann. Probab. 27 (4), 1851–1869 (1999)
Volkov, S.: Excited random walk on trees. Electr. J. Prob. paper 23, 2003
Zeitouni, O.: Random walks in random environment, XXXI Summer school in probability, St Flour, 2001. Lecture Notes in Mathematics 1837 193 – 312 (2004) (Springer)
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Most of this work was done while the author was Szegö Assistant Professor at Stanford University.
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Zerner, M. Multi-excited random walks on integers. Probab. Theory Relat. Fields 133, 98–122 (2005). https://doi.org/10.1007/s00440-004-0417-0
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DOI: https://doi.org/10.1007/s00440-004-0417-0