Non-homogeneous random walks on a half strip with generalized Lamperti drifts

Abstract

We study a Markov chain on $\mathbb{R}_+ \times S$, where $\mathbb{R}_+$ is the non-negative real numbers and $S$ is a finite set, in which when the $\mathbb{R}_+$-coordinate is large, the $S$-coordinate of the process is approximately Markov with stationary distribution $π_i$ on $S$. If $μ_i(x)$ is the mean drift of the $\mathbb{R}_+$-coordinate of the process at $(x,i) \in \mathbb{R}_+ \times S$, we study the case where $\sum_{i} π_i μ_i (x) \to 0$, which is the critical regime for the recurrence-transience phase transition. If $μ_i(x) \to 0$ for all $i$, it is natural to study the Lamperti case where $μ_i(x) = O(1/x)$; in that case the recurrence classification is known, but we prove new results on existence and non-existence of moments of return times. If $μ_i (x) \to d_i$ for $d_i \neq 0$ for at least some $i$, then it is natural to study the generalized Lamperti case where $μ_i (x) = d_i + O (1/x)$. By exploiting a transformation which maps the generalized Lamperti case to the Lamperti case, we obtain a recurrence classification and existence of moments results for the former. The generalized Lamperti case is seen to be more subtle, as the recurrence classification depends on correlation terms between the two coordinates of the process.