Conservative and Semiconservative Random Walks: Recurrence and Transience

Abstract

In the present paper we define conservative and semiconservative random walks in Z d and study different families of random walks. The family of symmetric random walks is one of the families of conservative random walks, and simple (Pólya) random walks are their representatives. The classification of random walks given in the present paper enables us to provide a new approach to random walks in Z d by reduction to birth-and-death processes. We construct nontrivial examples of recurrent random walks in Z d for any d ≥ 3 and transient random walks in Z 2 .

Correction to: Conservative and Semiconservative Random Walks: Recurrence and Transience

“…As n → ∞, asymptotic expansion (4.5) obtained in the proof of Lemma 4.2 in [1] guarantees its correctness. However, the corrected version of Lemma 4.1 requires more delicate arguments in the proofs of Lemma 4.2 and Theorem 4.13 in [1].…”

“…As n → ∞, asymptotic expansion (4.5) obtained in the proof of Lemma 4.2 in [1] guarantees its correctness. However, the corrected version of Lemma 4.1 requires more delicate arguments in the proofs of Lemma 4.2 and Theorem 4.13 in [1]. Specifically, in the proof of Lemma 4.2 instead of limit relation (4.6) we should study the cases d = 2 and d ≥ 3 separately in terms of the present formulation of Lemma 4.1.…”

“…Lemma 4. 1 Let the birth-and-death rates of a birth-and-death process be λ n and μ n all belonging to (0, ∞). Then, the birth-and-death process is transient if there exist c > 1 and a value n 0 such that for all n > n 0…”

“…for all large n and a small positive . In the proof of Theorem 4.13 in [1], we should take into account that for large n…”

Conditions for recurrence and transience for one family of random walks

“…Non-homogeneous random walks and their classification have been studied in book [19], where Lyapunov function methods for near-critical stochastic systems have been developed. Another new approach for classification of random walks through the concepts of conservative and semi-conservative random walks has been provided in [1]. In the present paper, we develop the methods related to [1], and the main contribution of the present paper is a closed form explicit asymptotic formula for the parameter that enables us to classify whether a random walk S t (a) is transient or recurrent for a quite general family of so-called nearest-neighborhood random walks {S t , A}.…”