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

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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 τα from a wedge with apex at the origin and interior half-angle α by a non-homogeneous random walk on the square lattice with mean drift at x of magnitude O(1/|x|) as |x| → ∞. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors (see arXiv:0910.1772) stated that τα< ∞ a.s. for any α (while for a stronger drift field τα is infinite with positive probability). Here we study the more difficult problem of the existence and non-existence of moments E[ταˢ], s>0. Assuming (in common with much of the literature) a uniform bound on the walk's increments, we show that for α< π/2 there exists s₀ ∈ (0,∞) such that E[ταˢ] is finite for s < s₀ but infinite for s > s₀; under specific assumptions on the drift field we show that we can attain E[ταˢ] = ∞ for any s > 1/2. We show that for α≤ π 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.