Angular Asymptotics for Multi-Dimensional Non-Homogeneous Random Walks with Asymptotically Zero Drift

#### I.M. MacPhee, M.V. Menshikov, A.R. Wade

2010, v.16, Issue 2, 351-388

ABSTRACT

We study the first exit time $\tau$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at $\bx \in \Z^d$ is of magnitude $O(\| \bx\|^{-1})$, we show that $\tau<\infty$ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude $\| \bx\|^{-\beta}$, $\beta \in (0,1)$, we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on $2$nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model.

Keywords: asymptotic direction,exit from cones,inhomogeneous randomwalk,perturbed random walk,random walk in random environment