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

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 $\pi_i$ on $S$. If $\mu_i(x)$ 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 $\sum_{i} \pi_i \mu_i (x) \to 0$, which is the critical regime for the recurrence-transience phase transition. If $\mu_i(x) \to 0$ for all $i$, it is natural to study the Lamperti case where $\mu_i(x) = O(1/x)$; in that case the recurrence classification is known, but we prove new results on existence and non-existence of moments of return times. If $\mu_i (x) \to d_i$ for $d_i \neq 0$ for at least some $i$, then it is natural to study the generalized Lamperti case where $\mu_i (x) = d_i + O (1/x)$. 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.