Exit times from cones for non-homogeneous random walk with asymptotically zero drift

We study the first exit time $\tau$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk on $\mathbb{Z}^2$ with mean drift that is asymptotically zero. Assuming bounded jumps and a form of weak isotropy, we give conditions for $\tau$ to be almost surely finite, and for the existence and non-existence of moments $\Exp [ \tau^p]$, $p>0$. Specifically, if the mean drift at $\bx \in \mathbb{Z}^2$ is of magnitude $O(\| \bx\|^{-1})$, then $\tau