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 τ α from a wedge with apex at the origin and interior half-angle α by a non-homogeneous random walk on ℤ2 with mean drift at x of magnitude O(∥x∥−1) as ∥x∥→∞. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors stated that τ α <∞ a.s. for any α. Here we study the more difficult problem of the existence and non-existence of moments \({\mathbb{E}}[ \tau_{\alpha}^{s}]\), s>0. Assuming a uniform bound on the walk’s increments, we show that for α<π/2 there exists s 0∈(0,∞) such that \({\mathbb{E}}[ \tau_{\alpha}^{s}]\) 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 \({\mathbb{E}}[ \tau_{\alpha}^{s}] = \infty\) for any s>1/2. We show that there is a phase transition between drifts of magnitude O(∥x∥−1) (the critical regime) and o(∥x∥−1) (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.