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

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 $\alpha \le \pi$ 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.

35 pages, 2 figures; most of the material in v1 (which had a different title) appears in revised form in v2 or the companion paper arXiv:0910.1772

References in corpus (3)

• Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains
• Gaussian estimates for spatially inhomogeneous random walks on ${\mathbf{Z}}^d$
• Rate of escape and central limit theorem for the supercritical Lamperti problem

Cited by in corpus (3)

• Random walks in cones
• Reflecting random walks in curvilinear wedges
• Passage-times for partially-homogeneous reflected random walks on the quadrant