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Consider a simple random walk on the integers with the following transitionmechanism. At each site $x$, the probability of jumping to the right is$\omega(x)\in[\frac12,1)$, until the first time the process jumps to the leftfrom site $x$, from which time onward the probability of jumping to the rightis $\frac12$. We investigate the transience/recurrence properties of thisprocess in both deterministic and stationary, ergodic environments$\{\omega(x)\}_{x\in Z}$. In deterministic environments, we also study thespeed of the process.

Transience/Recurrence and the speed of a one-dimensional random walk in a "have your cookie and eat it" environment