Markov chains with heavy-tailed increments and asymptotically zero drift

We study the recurrence/transience phase transition for Markov chains on ${\mathbb{R}}_{+}$, $\mathbb{R}$, and ${\mathbb{R}}^2$ whose increments have heavy tails with exponent in $(1,2)$ and asymptotically zero mean. This is the infinite-variance analogue of the classical Lamperti problem. On ${\mathbb{R}}_{+}$, for example, we show that if the tail of the positive increments is about $c{y}^{-\alpha }$ for an exponent $\alpha \in (1,2)$ and if the drift at $x$ is about $b{x}^{-\gamma }$, then the critical regime has $\gamma = \alpha -1$andrecurrence\mathrm{/}transienceisdeterminedbythesignof$b + c\pi $\text{cosec}$ (\pi \alpha)$\mathrm{.}On$$\mathbb{R}$$ we classify whether transience is directional or oscillatory, and extend an example of Rogozin & Foss to a class of transient martingales which oscillate between $\pm \mathrm{\infty }$. In addition to our recurrence/transience results, we also give sharp results on the existence/non-existence of moments of passage times.