We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on Z disturbed by cookies inducing a drift to the right of strength 1.
"Recurrence and transience of critical branching processes in random environment with immigration and an application to excited random walks." Adv. in Appl. Probab. 46 (3) 687 - 703, September 2014. https://doi.org/10.1239/aap/1409319555