Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts

Iain M. MacPhee, Mikhail V. Menshikov, and Andrew R. Wade

Journal of Theoretical Probability, 26, no. 1, March 2013, 1–30. DOI: 10.1007/s10959-012-0411-x

Abstract

We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time $\tau_\alpha$ from a wedge with apex at the origin and interior half-angle $\alpha$ by a non-homogeneous random walk on the square lattice with mean drift at $x$ of magnitude $O(1/|x|)$ as $|x| \to \infty$. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors (see arXiv:0910.1772) stated that $\tau_\alpha < \infty$ a.s. for any $\alpha$ (while for a stronger drift field $\tau_\alpha$ is infinite with positive probability). Here we study the more difficult problem of the existence and non-existence of moments $E[\tau_\alpha^s]$, $s>0$. Assuming (in common with much of the literature) a uniform bound on the walk's increments, we show that for $\alpha < \pi/2$ there exists $s_0 \in (0,\infty)$ such that $E[\tau_\alpha^s]$ is finite for $s < s_0$ but infinite for $s > s_0$; under specific assumptions on the drift field we show that we can attain $E[\tau_\alpha^s] = \infty$ for any $s > 1/2$. We show that for $\alpha \leq \pi$ there is a phase transition between drifts of magnitude $O(1/|x|)$ (the critical regime) and $o(1/|x|)$ (the subcritical regime). In the subcritical regime we obtain a non-homogeneous random walk analogue of a theorem for Brownian motion due to Spitzer, under considerably weaker conditions than those previously given (including work by Varopoulos) that assumed zero drift.