Persistent Random Walks. I. Recurrence Versus Transience

Abstract

We consider a walker on the line that at each step keeps the same direction with a probability which depends on the discrete time already spent in the direction the walker is currently moving. More precisely, the associated left-infinite sequence of jumps is supposed to be a Variable Length Markov Chain (VLMC) built from a probabilized context tree given by a double-infinite comb. These walks with memories of variable length can be seen as generalizations of Directionally Reinforced Random Walks (DRRW) introduced in [1, Mauldin & al., Adv. Math., 1996] in the sense that the persistence times are anisotropic. We give a complete characterization of the recurrence and the transience in terms of the probabilities to persist in the same direction or to switch. We point out that the underlying VLMC is not supposed to admit any stationary probability. Actually, the most fruitful situations emerge precisely when there is no such invariant distribution. In that case, the recurrent and the transient property are related to the behaviour of some embedded random walk with an undefined drift so that the asymptotic behaviour depends merely on the asymptotics of the probabilities of change of directions unlike the other case in which the criterion reduces to a drift condition. Finally, taking advantage of this flexibility, we give some (possibly random and lacunar) perturbations results and treat the case of more general probabilized context trees built by grafting subtrees onto the double-infinite comb.

Persistent Random Walks. II. Functional Scaling Limits

“…Furthermore, we deduce from (3.10) and (3.7) the following interpretation of the so-called mean drift m S -appearing here as a drift in probability -since for α P r1, 2s, the following holds lim nÑ8 S n n P " m S . Even if the latter convergence is a straightforward consequence of the almost sure convergence exposed in [1] under the assumption that the running times are both integrable, it is completely new otherwise.…”

“…The proof of the following proposition is omitted and follows from comparison results exposed in [1]. Proposition 3.1.…”

“…This paper is a continuation of [1] in which recurrence versus transience features of some PRWs are described. More specifically, we still consider a walker tS n u ně0 on Z, whose jumps are of unit size, and such that at each step it keeps the same direction (or switches) with a probability directly depending on the time already spent in the direction the walker is currently moving.…”

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We give a complete and unified description -under some stability assumptions -of the functional scaling limits associated with some persistent random walks for which the recurrent or transient type is studied in [1]. As a result, we highlight a phase transition phenomenon with respect to the memory.

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Recurrence of multidimensional persistent random walks. Fourier and series criteria

“…Inspired by [1], we produce a genuine counterexample to the conjecture of [2]. As for the one-dimensional situation studied in [3], it is easier for a persistent random walk than its skeleton to be recurrent but here the difference is extremely thin. These results are based on a surprisingly novel -to our knowledge -upper bound for the Lévy concentration function associated with symmetric distributions.…”