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

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

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 τα 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 τα<∞ a.s. for any α. Here we study the more difficult problem of the existence and non-existence of moments , s>0. Assuming a uniform bound on the walk’s increments, we show that for α<π/2 there exists s0∈(0,∞) such that TeX is finite for ss0; under specific assumptions on the drift field, we show that we can attain TeX 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.

Citation

MacPhee, I., Menshikov, M., & Wade, A. (2013). Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts. Journal of Theoretical Probability, 26(1), 1-30. https://doi.org/10.1007/s10959-012-0411-x

• Journal Article Type: Article
• Publication Date: Mar 1, 2013
• Deposit Date: Feb 6, 2013
• Publicly Available Date: Feb 14, 2013
• Journal: Journal of Theoretical Probability
• Print ISSN: 0894-9840
• Electronic ISSN: 1572-9230
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 26
• Issue: 1
• Pages: 1-30
• DOI: https://doi.org/10.1007/s10959-012-0411-x
• Public URL: https://durham-repository.worktribe.com/output/1468656