Perturbing transient random walk in a random environment with cookies of maximal strength

Abstract

We consider a left-transient random walk in a random environment on Z that will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.

Recurrence and transience of contractive autoregressive processes and related Markov chains

“…An application to random walks in random environments perturbed by cookies of maximal strength. We consider the same version of excited random walks in random environment as Bauernschubert in [Bau13]. Let ω = (ω x ) x∈Z be an i.i.d.…”

“…To the best of our knowledge there is at present no complete classification in simple terms of recurrence versus transience of these processes although this problem has been investigated for several decades, see [Pak75], [Pak79], [Kel92, Part I], [GM00, p. 1196], [ZG04], [Bau13] and the review below.…”

Null recurrence and transience of random difference equations in the contractive case

“…Further information on RDE and perpetuities can be found in the recent books [7] and [15]. If (2), (3), lim n→∞ Π n = 0 a.s. and I Q = ∞…”

Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

“…This paper complements [4]. In [4] a random walk in a random environment on Z which is transient to the left and disturbed by cookies of strength 1 to the right was considered.…”

“…This paper complements [4]. In [4] a random walk in a random environment on Z which is transient to the left and disturbed by cookies of strength 1 to the right was considered. As can be seen in [4], the study of special kinds of branching processes is essential to obtain results on recurrence and transience of these excited random walks.…”